{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0001","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"036930ebabeae81490556eb44c462cf5b3fa7e3c5a0e5208b7f3271d924d9f8e","family_id":"qf-mod35-crt-solve-pair","seed":202611045001,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,5],\"residues\":[0,1]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5] and aligned original residues=[0, 1] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 5], [0, 1], 6, 15, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 5], [0, 1], 6, 15, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0002","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a13743e510ec4c0394503ae163eaf1c37bc87be6651485001b473778edfa0a27","family_id":"qf-mod35-crt-solve-pair","seed":202611045002,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,7],\"residues\":[1,3]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 7] and aligned original residues=[1, 3] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 7], [1, 3], 10, 21, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 7], [1, 3], 10, 21, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0003","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28f20816fc3c26d411fa78018102e37a17e9645976b5120eb87b44d45fdb4ab4","family_id":"qf-mod35-crt-solve-pair","seed":202611045003,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,8],\"residues\":[2,5]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 8] and aligned original residues=[2, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 8], [2, 5], 5, 24, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 8], [2, 5], 5, 24, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0004","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9318254a3e5a5d13c9426f5f43730f022bc848e743db357e0bca45b9bf17e011","family_id":"qf-mod35-crt-solve-pair","seed":202611045004,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,10],\"residues\":[3,7]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 10] and aligned original residues=[3, 7] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 10], [0, 7], 27, 30, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 10], [0, 7], 27, 30, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0005","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30aa9228fdc8a7cc8c2c72b95cac2b92a474f3b02c620053c1d7d9de76e3ab38","family_id":"qf-mod35-crt-solve-pair","seed":202611045005,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,11],\"residues\":[4,9]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 11] and aligned original residues=[4, 9] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 11], [1, 9], 31, 33, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 11], [1, 9], 31, 33, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0006","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"807f960e43718d1a4dbaed0d3178cbd6e11d63128ac70640eb79aaeb5f1456b4","family_id":"qf-mod35-crt-solve-pair","seed":202611045006,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,13],\"residues\":[5,11]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 13] and aligned original residues=[5, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 13], [2, 11], 11, 39, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 13], [2, 11], 11, 39, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0007","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ef0c55d438101ae25ecae89cd1a10c3b524d75fc37127b05286295d800b240f","family_id":"qf-mod35-crt-solve-pair","seed":202611045007,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,14],\"residues\":[6,13]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 14] and aligned original residues=[6, 13] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 14], [0, 13], 27, 42, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 14], [0, 13], 27, 42, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0008","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e82677f4a6d363f9a8775c4b7044c50c27bf0f59f1a36fdcd83df778c8363213","family_id":"qf-mod35-crt-solve-pair","seed":202611045008,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,16],\"residues\":[7,15]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 16] and aligned original residues=[7, 15] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 16], [1, 15], 31, 48, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 16], [1, 15], 31, 48, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0009","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07fced59ab2e37c29022012afedffe2814f2c3a8359bce80f949515351c8afeb","family_id":"qf-mod35-crt-solve-pair","seed":202611045009,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,17],\"residues\":[8,17]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 17] and aligned original residues=[8, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 17], [2, 0], 17, 51, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 17], [2, 0], 17, 51, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0010","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad655f8201375f38f19551b3ce4ce79227d89f051a8b50dfda2c83a7af92e5d9","family_id":"qf-mod35-crt-solve-pair","seed":202611045010,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,19],\"residues\":[9,19]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 19] and aligned original residues=[9, 19] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 19], [0, 0], 0, 57, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 19], [0, 0], 0, 57, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0011","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1da3780bab33fb8170ccd26343a76038be6be7e01dba96d4ec69a2946c428a60","family_id":"qf-mod35-crt-solve-pair","seed":202611045011,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,20],\"residues\":[10,21]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 20] and aligned original residues=[10, 21] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 20], [1, 1], 1, 60, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 20], [1, 1], 1, 60, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0012","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d1a788ac9748a03ebb24e5b7f1684929ea5ba60969494760edc82963011d508","family_id":"qf-mod35-crt-solve-pair","seed":202611045012,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,22],\"residues\":[11,23]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 22] and aligned original residues=[11, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 22], [2, 1], 23, 66, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 22], [2, 1], 23, 66, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0013","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b974f1e62975c76dc0f93552e2960f408b5209cb114df06edf38b0fa966febf","family_id":"qf-mod35-crt-solve-pair","seed":202611045013,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,23],\"residues\":[12,25]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 23] and aligned original residues=[12, 25] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 23], [0, 2], 48, 69, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 23], [0, 2], 48, 69, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0014","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4ae89f206dd5763fee6720be9b604dc5572ef87787faa7b6ca089e70bcd30b","family_id":"qf-mod35-crt-solve-pair","seed":202611045014,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,25],\"residues\":[13,27]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 25] and aligned original residues=[13, 27] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 25], [1, 2], 52, 75, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 25], [1, 2], 52, 75, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0015","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0075944cf1ea4416020383883854d3b16fbd624ecaa4c939e684d22ed546591a","family_id":"qf-mod35-crt-solve-pair","seed":202611045015,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,26],\"residues\":[14,29]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 26] and aligned original residues=[14, 29] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 26], [2, 3], 29, 78, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 26], [2, 3], 29, 78, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0016","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf580b7b1d579165d79beb0b0aa543ee28b5a72d676e525f31c0168072f7efd9","family_id":"qf-mod35-crt-solve-pair","seed":202611045016,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,28],\"residues\":[15,31]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 28] and aligned original residues=[15, 31] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 28], [0, 3], 3, 84, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 28], [0, 3], 3, 84, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0017","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aab9f1c2e479fb0a733a77564804bcf3e5df851057d2dab30565dd99ea71059","family_id":"qf-mod35-crt-solve-pair","seed":202611045017,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,29],\"residues\":[16,33]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 29] and aligned original residues=[16, 33] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 29], [1, 4], 4, 87, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 29], [1, 4], 4, 87, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0018","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adc6e1b4e5711a46e9698c430021cf911a4c0ce1c64e2e165ea851eedc2baefd","family_id":"qf-mod35-crt-solve-pair","seed":202611045018,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,31],\"residues\":[17,35]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 31] and aligned original residues=[17, 35] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 31], [2, 4], 35, 93, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 31], [2, 4], 35, 93, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0019","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5a8eaba2d37e423386c28c7b83508fbd517c57560728a7ce56f8a93e592510","family_id":"qf-mod35-crt-solve-pair","seed":202611045019,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,32],\"residues\":[18,37]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 32] and aligned original residues=[18, 37] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 32], [0, 5], 69, 96, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 32], [0, 5], 69, 96, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0020","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f62f2fd77444144801cb31b8b9456fb40ea2399f657f52b8a5a1b4c0bb8287a","family_id":"qf-mod35-crt-solve-pair","seed":202611045020,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,34],\"residues\":[19,39]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 34] and aligned original residues=[19, 39] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 34], [1, 5], 73, 102, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 34], [1, 5], 73, 102, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0021","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5f937789e4600d0ed7e568caa2950c1a8812aa5639e203dbae883cd66a24a86","family_id":"qf-mod35-crt-solve-pair","seed":202611045021,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,35],\"residues\":[20,41]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 35] and aligned original residues=[20, 41] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 35], [2, 6], 41, 105, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 35], [2, 6], 41, 105, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0022","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3de10ce8b06d3e99162908d83e0839e12c3042c33e778260dc54d45a2dd419bc","family_id":"qf-mod35-crt-solve-pair","seed":202611045022,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,37],\"residues\":[21,43]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 37] and aligned original residues=[21, 43] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 37], [0, 6], 6, 111, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 37], [0, 6], 6, 111, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0023","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93e075c5137a70bd5e09880279c25dcfde80a88cfdfb2177eb2a034c204f0854","family_id":"qf-mod35-crt-solve-pair","seed":202611045023,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,38],\"residues\":[22,45]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 38] and aligned original residues=[22, 45] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 38], [1, 7], 7, 114, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 38], [1, 7], 7, 114, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0024","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb857f3e207a05bed99150320905cb5676b7b37704925ed6b653aa21580f7da0","family_id":"qf-mod35-crt-solve-pair","seed":202611045024,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,40],\"residues\":[23,47]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 40] and aligned original residues=[23, 47] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 40], [2, 7], 47, 120, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 40], [2, 7], 47, 120, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0025","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"679a4cf0c9a79436b58117a2a5e2d97764753568e726f1409d7f926fe1b96413","family_id":"qf-mod35-crt-solve-pair","seed":202611045025,"semantic_template":"qf-mod35-crt-solve-pair/{\"moduli\":[3,41],\"residues\":[24,49]}"},"fields":{"title":"Solve a Pairwise-Coprime Pair after Normalizing Both Residues","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 41] and aligned original residues=[24, 49] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).","answer":{"kind":"expression","value":"([3, 41], [0, 8], 90, 123, Yes)","explanation":"Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 41], [0, 8], 90, 123, Yes)."},"skills":["Solve a Pairwise-Coprime Pair after Normalizing Both Residues"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0026","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89d4bf0a0832ca5a85a354fad05481d25bee0d9fcebfdb4181411cdef3bacf7","family_id":"qf-mod35-crt-solve-system","seed":202611045026,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,7],\"residues\":[0,1,2]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 7] and aligned original residues=[0, 1, 2] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 7], [0, 1, 2], 51, 105, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 7], [0, 1, 2], 51, 105, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0027","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82680bb687db0e7225aaa5e7fbe1a7cc8ba6a95ab290f8c83739a996ab06d161","family_id":"qf-mod35-crt-solve-system","seed":202611045027,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,8],\"residues\":[1,3,5]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 8] and aligned original residues=[1, 3, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 8], [1, 3, 5], 13, 120, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 8], [1, 3, 5], 13, 120, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0028","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38017188cd82c8bcdf3b79b26e59d698dd17e1cee3f5c18e24069bd28daad789","family_id":"qf-mod35-crt-solve-system","seed":202611045028,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,11],\"residues\":[2,5,8]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 11] and aligned original residues=[2, 5, 8] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 11], [2, 0, 8], 140, 165, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 11], [2, 0, 8], 140, 165, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0029","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccc6fcb8f8f6772174d237aa4a6d4db79b4d69a647f1b652de5b5f96dec4d930","family_id":"qf-mod35-crt-solve-system","seed":202611045029,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,13],\"residues\":[3,7,11]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 13] and aligned original residues=[3, 7, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 13], [0, 2, 11], 102, 195, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 13], [0, 2, 11], 102, 195, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0030","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b224537f061e81d602e5feb080a85cc83b65c1a88bfaaab97049cd96a9f19fe","family_id":"qf-mod35-crt-solve-system","seed":202611045030,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,14],\"residues\":[4,9,14]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 14] and aligned original residues=[4, 9, 14] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 14], [1, 4, 0], 154, 210, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 14], [1, 4, 0], 154, 210, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0031","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbf63f3f3d002142992027f86a03cf7ffb605454c829820d2ecc2a1942d8599a","family_id":"qf-mod35-crt-solve-system","seed":202611045031,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,16],\"residues\":[5,11,17]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 16] and aligned original residues=[5, 11, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 16], [2, 1, 1], 161, 240, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 16], [2, 1, 1], 161, 240, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0032","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a35b22b3bdf10120577f967b19c524fff0bde8a62ccd03c89ff24e758992d0","family_id":"qf-mod35-crt-solve-system","seed":202611045032,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,17],\"residues\":[6,13,20]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 17] and aligned original residues=[6, 13, 20] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 17], [0, 3, 3], 3, 255, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 17], [0, 3, 3], 3, 255, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0033","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f8dc0db7c10599c4656448c489bb29567cf9985088990445adbf2b7fa4243c1","family_id":"qf-mod35-crt-solve-system","seed":202611045033,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,19],\"residues\":[7,15,23]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 19] and aligned original residues=[7, 15, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 19], [1, 0, 4], 175, 285, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 19], [1, 0, 4], 175, 285, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0034","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e61c889a6b3b6f2e9a826a3c48e8130e0983503f408ae1496e2f0f9047f938f","family_id":"qf-mod35-crt-solve-system","seed":202611045034,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,22],\"residues\":[8,17,26]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 22] and aligned original residues=[8, 17, 26] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 22], [2, 2, 4], 92, 330, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 22], [2, 2, 4], 92, 330, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0035","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b32f414f20d9ec9af2aa299ccae3905960cd98d871e28291316d476ef628461c","family_id":"qf-mod35-crt-solve-system","seed":202611045035,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,23],\"residues\":[9,19,29]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 23] and aligned original residues=[9, 19, 29] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 23], [0, 4, 6], 144, 345, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 23], [0, 4, 6], 144, 345, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0036","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560eb780ed1905293eae039f5d85428077f84498f7f48c1429e2bdda8bbc9d86","family_id":"qf-mod35-crt-solve-system","seed":202611045036,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,26],\"residues\":[10,21,32]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 26] and aligned original residues=[10, 21, 32] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 26], [1, 1, 6], 136, 390, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 26], [1, 1, 6], 136, 390, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0037","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a58865ba9c00d15964932c4ff8bf2e76805d23074ce017b293e3cb43558027be","family_id":"qf-mod35-crt-solve-system","seed":202611045037,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,28],\"residues\":[11,23,35]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 28] and aligned original residues=[11, 23, 35] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 28], [2, 3, 7], 203, 420, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 28], [2, 3, 7], 203, 420, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0038","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1af0d43bb2f9491a513486153c564371ef06e67809e8a3b5e9c52ad0ef170465","family_id":"qf-mod35-crt-solve-system","seed":202611045038,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,29],\"residues\":[12,25,38]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 29] and aligned original residues=[12, 25, 38] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 29], [0, 0, 9], 270, 435, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 29], [0, 0, 9], 270, 435, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0039","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1296d26d72b0768b610d81dadd0fe7d26e3638a76f63d65c51409b0a27975c3","family_id":"qf-mod35-crt-solve-system","seed":202611045039,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,31],\"residues\":[13,27,41]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 31] and aligned original residues=[13, 27, 41] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 31], [1, 2, 10], 382, 465, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 31], [1, 2, 10], 382, 465, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0040","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e42131b18e000c1d5cbe9cf396bcad21c1232301663f33918be70120a00b066","family_id":"qf-mod35-crt-solve-system","seed":202611045040,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,32],\"residues\":[14,29,44]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 32] and aligned original residues=[14, 29, 44] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 32], [2, 4, 12], 44, 480, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 32], [2, 4, 12], 44, 480, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0041","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df6af77c3df1e407fc1ea97df5bbaf717de92404f335d5206e34824c12deb1cc","family_id":"qf-mod35-crt-solve-system","seed":202611045041,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,34],\"residues\":[15,31,47]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 34] and aligned original residues=[15, 31, 47] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 34], [0, 1, 13], 81, 510, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 34], [0, 1, 13], 81, 510, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0042","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cde0ef92c097a5afbc4f0e3e571c472fe9c6a926543de493978ff1c130a3433e","family_id":"qf-mod35-crt-solve-system","seed":202611045042,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,37],\"residues\":[16,33,50]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 37] and aligned original residues=[16, 33, 50] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 37], [1, 3, 13], 13, 555, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 37], [1, 3, 13], 13, 555, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0043","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"514ed445db942ebd09911349cf91297e3140eda3fedac9670bca6ef8654dfb63","family_id":"qf-mod35-crt-solve-system","seed":202611045043,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,38],\"residues\":[17,35,53]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 38] and aligned original residues=[17, 35, 53] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 38], [2, 0, 15], 395, 570, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 38], [2, 0, 15], 395, 570, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0044","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79b8b78a46a9f70718b560c3119635f54afe5333171f0dbf27edfad0bc975e45","family_id":"qf-mod35-crt-solve-system","seed":202611045044,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,41],\"residues\":[18,37,56]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 41] and aligned original residues=[18, 37, 56] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 41], [0, 2, 15], 507, 615, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 41], [0, 2, 15], 507, 615, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0045","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5adb243d9e835e4e5c9bbce9eab5ab24241719b40a7dc07c88ad246b7b48c38b","family_id":"qf-mod35-crt-solve-system","seed":202611045045,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,43],\"residues\":[19,39,59]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 43] and aligned original residues=[19, 39, 59] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 43], [1, 4, 16], 274, 645, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 43], [1, 4, 16], 274, 645, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0046","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8e9ee9309c85d6d7872925659540945a7a7d486cbea5fc93a083c1134333dab","family_id":"qf-mod35-crt-solve-system","seed":202611045046,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,44],\"residues\":[20,41,62]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 44] and aligned original residues=[20, 41, 62] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 44], [2, 1, 18], 326, 660, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 44], [2, 1, 18], 326, 660, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0047","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd44a653bf84a565ebebd3e873f84dc5848892479a320a455b6dc4b7aecd907","family_id":"qf-mod35-crt-solve-system","seed":202611045047,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,46],\"residues\":[21,43,65]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 46] and aligned original residues=[21, 43, 65] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 46], [0, 3, 19], 663, 690, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 46], [0, 3, 19], 663, 690, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0048","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceb38a1d2870cb171bb0eb5978e437893bdc2d147c51a6f70b84c7663b44b147","family_id":"qf-mod35-crt-solve-system","seed":202611045048,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,47],\"residues\":[22,45,68]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 47] and aligned original residues=[22, 45, 68] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 47], [1, 0, 21], 115, 705, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 47], [1, 0, 21], 115, 705, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0049","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75b39f18f011a892bbc19d35a05e4bcafd0cf42b83cf4d16fa8b13fb4dd39db2","family_id":"qf-mod35-crt-solve-system","seed":202611045049,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,49],\"residues\":[23,47,71]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 49] and aligned original residues=[23, 47, 71] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 49], [2, 2, 22], 512, 735, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 49], [2, 2, 22], 512, 735, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0050","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70644cd2fb1f6912b6d96641e6efd7dc64f4cdfc75cfc4de6a4f3b91ceb52dd6","family_id":"qf-mod35-crt-solve-system","seed":202611045050,"semantic_template":"qf-mod35-crt-solve-system/{\"moduli\":[3,5,52],\"residues\":[24,49,74]}"},"fields":{"title":"Every Congruence Must Pass in the Same CRT System","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5, 52] and aligned original residues=[24, 49, 74] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).","answer":{"kind":"expression","value":"([3, 5, 52], [0, 4, 22], 594, 780, [Yes, Yes, Yes], Yes)","explanation":"Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 52], [0, 4, 22], 594, 780, [Yes, Yes, Yes], Yes)."},"skills":["Every Congruence Must Pass in the Same CRT System"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0051","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5be6699cc7e42d70a0a69433bf5dc64f3479d9fedf317d49acc04020bee9fca7","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045051,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,5],\"other\":21,\"residues\":[0,1]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 5] and aligned original residues=[0, 1] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=21. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(6, 21, 15, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (6, 21, 15, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0052","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a1a01d54b7c344f228b878b6c2fb8da59b4f98a913374604aa69069d6b5004d","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045052,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,7],\"other\":31,\"residues\":[1,3]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 7] and aligned original residues=[1, 3] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=31. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(10, 31, 21, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (10, 31, 21, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0053","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ded6aff93763cfa485cb24c457f735f63cf640af8e382863e41ed4af9dd00bd","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045053,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,8],\"other\":29,\"residues\":[2,5]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 8] and aligned original residues=[2, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=29. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(5, 29, 24, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (5, 29, 24, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0054","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63cc9ceecb036b88c5cccb26ec5ac8d10e83bd6cd7582331dfefd2b0575c8952","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045054,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,10],\"other\":57,\"residues\":[3,7]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 10] and aligned original residues=[3, 7] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=57. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(27, 57, 30, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (27, 57, 30, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0055","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"160a74667838551ee5c166be3026826c01ca8e7e69212c3fdaaf91dc8e6a1c00","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045055,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,11],\"other\":64,\"residues\":[4,9]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 11] and aligned original residues=[4, 9] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=64. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(31, 64, 33, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (31, 64, 33, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0056","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1fdc618586b297f4b94fff0b73075fc3143eead1419192986af9f38bcfb77c","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045056,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,13],\"other\":50,\"residues\":[5,11]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 13] and aligned original residues=[5, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=50. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(11, 50, 39, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (11, 50, 39, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0057","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"416398877bd54ec468d81d999baa52f2eaa55bd2493ef730c79c3a14a4b125ad","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045057,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,14],\"other\":69,\"residues\":[6,13]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 14] and aligned original residues=[6, 13] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=69. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(27, 69, 42, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (27, 69, 42, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0058","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc16088167af110bed58f22d81f84738a657523b5190a6108e08786787de6813","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045058,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,16],\"other\":79,\"residues\":[7,15]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 16] and aligned original residues=[7, 15] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=79. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(31, 79, 48, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (31, 79, 48, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0059","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d51186fd5447a4130ea8a0cf013dc2c03e60f646e3803df26913aa80c9d76e","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045059,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,17],\"other\":68,\"residues\":[8,17]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 17] and aligned original residues=[8, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=68. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(17, 68, 51, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (17, 68, 51, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0060","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fd9a46508eec1811ce775dc6c7657f9ccbd52088ec02a82959dfc3858b8cc92","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045060,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,19],\"other\":57,\"residues\":[9,19]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 19] and aligned original residues=[9, 19] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=57. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(0, 57, 57, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (0, 57, 57, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0061","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6170ad72619fcc514c7b8432be8f74d1079b75189c818e2548c25e15f685f21a","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045061,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,20],\"other\":61,\"residues\":[10,21]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 20] and aligned original residues=[10, 21] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=61. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(1, 61, 60, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (1, 61, 60, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0062","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3926a16652ffc0d465ac211c915aa7e3a1899c18f31823b312a42c0cd934e80","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045062,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,22],\"other\":89,\"residues\":[11,23]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 22] and aligned original residues=[11, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=89. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(23, 89, 66, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (23, 89, 66, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0063","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aa48f11777c7aab8fecce845515d56b0456932add4e54a9656fcbef52e7be29","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045063,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,23],\"other\":117,\"residues\":[12,25]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 23] and aligned original residues=[12, 25] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=117. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(48, 117, 69, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (48, 117, 69, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0064","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a87d704a55ba69597d32f4b6010361bba6f302c9b9d33f3fecb496ac964f5ab2","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045064,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,25],\"other\":127,\"residues\":[13,27]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 25] and aligned original residues=[13, 27] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=127. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(52, 127, 75, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (52, 127, 75, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0065","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e51c407ba34bd5f53026dd6598b652da735246af63e1a760079a68a5fba9386","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045065,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,26],\"other\":107,\"residues\":[14,29]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 26] and aligned original residues=[14, 29] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=107. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(29, 107, 78, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (29, 107, 78, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0066","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96c6d89cfbcac55718b71fbffe9711e3b91740373d58f2f13e260b8f01f25bb0","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045066,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,28],\"other\":87,\"residues\":[15,31]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 28] and aligned original residues=[15, 31] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=87. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(3, 87, 84, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (3, 87, 84, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0067","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4501d1055f503ecfac68d66d2d353f1420277148326501916524fb04dad4df00","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045067,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,29],\"other\":91,\"residues\":[16,33]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 29] and aligned original residues=[16, 33] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=91. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(4, 91, 87, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (4, 91, 87, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0068","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e9df1bc6198a015cdcfc7ade149356627b1280c030a65332a0d2e55fe421bf9","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045068,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,31],\"other\":128,\"residues\":[17,35]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 31] and aligned original residues=[17, 35] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=128. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(35, 128, 93, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (35, 128, 93, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0069","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"300a6428a0dcc5ab48fc520c5e375b93ea4497871068a6ad8258e74380d3e820","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045069,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,32],\"other\":165,\"residues\":[18,37]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 32] and aligned original residues=[18, 37] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=165. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(69, 165, 96, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (69, 165, 96, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0070","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e6e3d37379a23afd8212666b35f60651ab643c7dec29f057e0d684f01f8f544","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045070,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,34],\"other\":175,\"residues\":[19,39]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 34] and aligned original residues=[19, 39] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=175. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(73, 175, 102, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (73, 175, 102, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0071","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af949d1bd4eb1d35bb0c4ed1bb762b1c565c14a08f7ac97ee91546f9d467998","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045071,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,35],\"other\":146,\"residues\":[20,41]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 35] and aligned original residues=[20, 41] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=146. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(41, 146, 105, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (41, 146, 105, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0072","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ae4f99ed94f28b01753923a8fd027f57a1bf741febc60da0719d9dedf103687","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045072,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,37],\"other\":117,\"residues\":[21,43]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 37] and aligned original residues=[21, 43] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=117. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(6, 117, 111, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (6, 117, 111, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0073","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a380a5a9b6183ffde7003202fdaa37e9378096c339b962d9ab9ec75040087873","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045073,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,38],\"other\":121,\"residues\":[22,45]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 38] and aligned original residues=[22, 45] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=121. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(7, 121, 114, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (7, 121, 114, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0074","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be1b5fc2b2bf5fe3b3a834808f9791343eb845bdb5455bbd4bd73e408e6d6ebe","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045074,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,40],\"other\":167,\"residues\":[23,47]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 40] and aligned original residues=[23, 47] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=167. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(47, 167, 120, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (47, 167, 120, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0075","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9795487288bedc5f4dc1842053382d8edbb1810e0e5e6cb5f3dfecf2f417a6a1","family_id":"qf-mod35-crt-uniqueness-mod-product","seed":202611045075,"semantic_template":"qf-mod35-crt-uniqueness-mod-product/{\"moduli\":[3,41],\"other\":213,\"residues\":[24,49]}"},"fields":{"title":"Uniqueness Is a Residue-Class Claim, Not a Unique Integer","stem":"Use the complete integer congruence system with positive distinct moduli=[3, 41] and aligned original residues=[24, 49] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=213. Give (solution, other, product, same_residue_class, boundary).","answer":{"kind":"expression","value":"(90, 213, 123, Yes, unique_modulo_product_not_unique_integer)","explanation":"Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (90, 213, 123, Yes, unique_modulo_product_not_unique_integer)."},"skills":["Uniqueness Is a Residue-Class Claim, Not a Unique Integer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0076","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa3ce4b811da208a3ce81409dade3af5855e884e201debb7574095410e8c6211","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045076,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[4,6],\"residues\":[1,3]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[4, 6] and aligned original residues=[1, 3] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([4, 6], [1, 3], Yes, 9, 12, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([4, 6], [1, 3], Yes, 9, 12, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0077","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"489f6bf8fcc1809b5b135a4a2d00a5eb355450c96d6fd783b5050437109cbc2e","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045077,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[6,9],\"residues\":[1,4]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[6, 9] and aligned original residues=[1, 4] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([6, 9], [1, 4], Yes, 13, 18, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([6, 9], [1, 4], Yes, 13, 18, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0078","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f2475ec341b42b7e51baa010851014448e730e6fec35c7b40d2f03f87cf1533","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045078,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[8,12],\"residues\":[1,5]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[8, 12] and aligned original residues=[1, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([8, 12], [1, 5], Yes, 17, 24, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([8, 12], [1, 5], Yes, 17, 24, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0079","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b326d623e974cfcc079688684be840101c8c9703891233c06a82b9440a1db84","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045079,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[10,15],\"residues\":[1,6]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[10, 15] and aligned original residues=[1, 6] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([10, 15], [1, 6], Yes, 21, 30, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([10, 15], [1, 6], Yes, 21, 30, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0080","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2c247f2ef57bc97af90f867093599aa54da1eb404392d2fbfadf70c7eec4bd5","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045080,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[12,18],\"residues\":[1,7]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[12, 18] and aligned original residues=[1, 7] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([12, 18], [1, 7], Yes, 25, 36, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([12, 18], [1, 7], Yes, 25, 36, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0081","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91296f103d1245e2d26c9f8094919281f1532fcb31108a0e976de9407cfbaa71","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045081,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[14,21],\"residues\":[1,8]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[14, 21] and aligned original residues=[1, 8] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([14, 21], [1, 8], Yes, 29, 42, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([14, 21], [1, 8], Yes, 29, 42, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0082","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab4115fed973a77e207fbd64e0852100f51b290fb3c2c2ad058a904ad3f35264","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045082,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[16,24],\"residues\":[1,9]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[16, 24] and aligned original residues=[1, 9] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([16, 24], [1, 9], Yes, 33, 48, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([16, 24], [1, 9], Yes, 33, 48, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0083","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bfea98f4079cee983d103babb4353509537ef5163a0a27031c21d894fbdd7ae","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045083,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[18,27],\"residues\":[1,10]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[18, 27] and aligned original residues=[1, 10] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([18, 27], [1, 10], Yes, 37, 54, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([18, 27], [1, 10], Yes, 37, 54, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0084","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b00877c8a038f9e426ffc539dab1d2e63eacf6ec44d74052d9366eaf86ff95","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045084,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[20,30],\"residues\":[1,11]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[20, 30] and aligned original residues=[1, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([20, 30], [1, 11], Yes, 41, 60, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([20, 30], [1, 11], Yes, 41, 60, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0085","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda944122df2d4b7508fc5732e2eea8e5ebf562a7d09baed6eaba720e94fd76b","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045085,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[22,33],\"residues\":[1,12]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[22, 33] and aligned original residues=[1, 12] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([22, 33], [1, 12], Yes, 45, 66, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([22, 33], [1, 12], Yes, 45, 66, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0086","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"149cc8648f3280847a0a2a65ca00138134ecd7082ef20f32d55563eb0a08daf0","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045086,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[24,36],\"residues\":[1,13]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[24, 36] and aligned original residues=[1, 13] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([24, 36], [1, 13], Yes, 49, 72, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([24, 36], [1, 13], Yes, 49, 72, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0087","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc776880b60766975bec3af041c9b0dc08ea0d786b74120e8b327a5fdc6bb38e","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045087,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[26,39],\"residues\":[1,14]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[26, 39] and aligned original residues=[1, 14] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([26, 39], [1, 14], Yes, 53, 78, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([26, 39], [1, 14], Yes, 53, 78, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0088","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b444bbe2d8c249fe2c195fb1e9ebdce4cf4bac26067579df15ba48ffa997909e","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045088,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[28,42],\"residues\":[1,15]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[28, 42] and aligned original residues=[1, 15] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([28, 42], [1, 15], Yes, 57, 84, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([28, 42], [1, 15], Yes, 57, 84, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0089","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f40fe4594c646b5ada1864225ac78c69f4f8871345010b113fa98a7573ea49c","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045089,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[30,45],\"residues\":[1,16]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[30, 45] and aligned original residues=[1, 16] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([30, 45], [1, 16], Yes, 61, 90, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([30, 45], [1, 16], Yes, 61, 90, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0090","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5091c9bb9c3c51c54d219185f3fd34eeaae50f180fde3cdf55e5ff0164fce7e7","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045090,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[32,48],\"residues\":[1,17]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[32, 48] and aligned original residues=[1, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([32, 48], [1, 17], Yes, 65, 96, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([32, 48], [1, 17], Yes, 65, 96, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0091","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce56bea1767b5fcb05ee795a17842fcc91491fe18347d095a2677fad56a205dc","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045091,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[34,51],\"residues\":[1,18]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[34, 51] and aligned original residues=[1, 18] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([34, 51], [1, 18], Yes, 69, 102, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([34, 51], [1, 18], Yes, 69, 102, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0092","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb3af272a7c1ba2dce66964422d8b994c23b845279e1cdb2dedbf235f5a1b686","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045092,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[36,54],\"residues\":[1,19]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[36, 54] and aligned original residues=[1, 19] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([36, 54], [1, 19], Yes, 73, 108, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([36, 54], [1, 19], Yes, 73, 108, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0093","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c6164cfeebecc9744053dc948a5ab857ba6561e8ac0ca518406140e8da99442","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045093,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[38,57],\"residues\":[1,20]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[38, 57] and aligned original residues=[1, 20] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([38, 57], [1, 20], Yes, 77, 114, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([38, 57], [1, 20], Yes, 77, 114, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0094","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92ed74b1918fc0567b34aaeacb637e79e256b7762b0825def3223423e356f0a4","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045094,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[40,60],\"residues\":[1,21]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[40, 60] and aligned original residues=[1, 21] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([40, 60], [1, 21], Yes, 81, 120, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([40, 60], [1, 21], Yes, 81, 120, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0095","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc3fc43bd9bbaef430909f38be46d487cc9ec0a0f8b36740d8dc9d8928f4b9ec","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045095,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[42,63],\"residues\":[1,22]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[42, 63] and aligned original residues=[1, 22] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([42, 63], [1, 22], Yes, 85, 126, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([42, 63], [1, 22], Yes, 85, 126, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0096","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be8819033f6d31e6f8e4445577717ad8a1185cbb8db5970d7f9949b1b4048379","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045096,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[44,66],\"residues\":[1,23]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[44, 66] and aligned original residues=[1, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([44, 66], [1, 23], Yes, 89, 132, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([44, 66], [1, 23], Yes, 89, 132, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0097","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"997271aaf3b62baf0776c3f4d3fb44f2e7f8c19f836a83b18c0a1297ac3a89ff","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045097,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[46,69],\"residues\":[1,24]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[46, 69] and aligned original residues=[1, 24] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([46, 69], [1, 24], Yes, 93, 138, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([46, 69], [1, 24], Yes, 93, 138, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0098","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ebcd4cd5e048173274c8a2fd4e71698a42233f66f9b1d669c81d552be289db4","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045098,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[48,72],\"residues\":[1,25]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[48, 72] and aligned original residues=[1, 25] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([48, 72], [1, 25], Yes, 97, 144, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([48, 72], [1, 25], Yes, 97, 144, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0099","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6900db4b78d5de7dc1bc99ff6dc45294f581bf174dfebfd7243acff5c072fe87","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045099,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[50,75],\"residues\":[1,26]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[50, 75] and aligned original residues=[1, 26] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([50, 75], [1, 26], Yes, 101, 150, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([50, 75], [1, 26], Yes, 101, 150, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r05-chinese-remainder-theorem-0100","batch_id":"batch-20261014-advance-20261104-r05-chinese-remainder-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810cad6f412d6f6100ee119e8afff4fd6dbb33d4cbbad0a01892c8cef93dc6e1","family_id":"qf-mod35-crt-noncoprime-boundary","seed":202611045100,"semantic_template":"qf-mod35-crt-noncoprime-boundary/{\"moduli\":[52,78],\"residues\":[1,27]}"},"fields":{"title":"A Noncoprime System Can Be Compatible or Incompatible","stem":"Use the complete integer congruence system with positive distinct moduli=[52, 78] and aligned original residues=[1, 27] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).","answer":{"kind":"expression","value":"([52, 78], [1, 27], Yes, 105, 156, noncoprime_not_automatically_inconsistent)","explanation":"Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([52, 78], [1, 27], Yes, 105, 156, noncoprime_not_automatically_inconsistent)."},"skills":["A Noncoprime System Can Be Compatible or Incompatible"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}