{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0001","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edb0a7293b6ea087e121d8649ad15538413c0b2bc750310b41daab395aaba54f","family_id":"qf-mod47-morder-qualification","seed":202611167001,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":3}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=3 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(3, 2, 2, 1, 2, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (3, 2, 2, 1, 2, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0002","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f91cc026dd6202ef72acc80b5bee0eaaae6940168579988d9f9780eaefb04f","family_id":"qf-mod47-morder-qualification","seed":202611167002,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":4}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(4, 3, 3, 1, 2, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (4, 3, 3, 1, 2, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0003","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfccdc04a766a9b7329713b6aba24e7de86855f867c39065cf3eb8464c812d1f","family_id":"qf-mod47-morder-qualification","seed":202611167003,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":5}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=5 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(5, 2, 2, 1, 4, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (5, 2, 2, 1, 4, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0004","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b01aa2fbc52787b0a97f1c1153e49a6f062d74bb16f3c67c6404a54c880922bf","family_id":"qf-mod47-morder-qualification","seed":202611167004,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":6}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=6 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(6, 5, 5, 1, 2, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (6, 5, 5, 1, 2, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0005","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"debaea77690dde428884ca98e859d5ddd382e14cdee2bbe7985981336c5fcb4f","family_id":"qf-mod47-morder-qualification","seed":202611167005,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":7}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=7 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(7, 2, 2, 1, 6, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (7, 2, 2, 1, 6, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0006","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"216bdd608c377d0068a328a490597501067faa4a6245a4e91c1f41db14559570","family_id":"qf-mod47-morder-qualification","seed":202611167006,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":8}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=8 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(8, 3, 3, 1, 4, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (8, 3, 3, 1, 4, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0007","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a3e1475789a43e774020b092307ccccf8264e5940e5a38f490398f0fddcc069","family_id":"qf-mod47-morder-qualification","seed":202611167007,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":9}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=9 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(9, 2, 2, 1, 6, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (9, 2, 2, 1, 6, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0008","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83d4019799e468f2d440cee520a3542dfcbbc9f29df7475e261b5fc48174ba4","family_id":"qf-mod47-morder-qualification","seed":202611167008,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":10}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=10 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(10, 3, 3, 1, 4, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (10, 3, 3, 1, 4, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0009","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9038677c7a3262eb931b69e7f14925c1275927d806e136a1e5c7ad19b938e551","family_id":"qf-mod47-morder-qualification","seed":202611167009,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":11}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=11 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(11, 2, 2, 1, 10, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (11, 2, 2, 1, 10, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0010","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"222802cf3acd7d8e5d29a2a5e311c0f3427cdc7e2f70c4027e52c240fb4c18b1","family_id":"qf-mod47-morder-qualification","seed":202611167010,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":12}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=12 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(12, 5, 5, 1, 4, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (12, 5, 5, 1, 4, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0011","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cbe4fda92b6d298490ac61e54689358552b7099eb128a03b6732e25939de662","family_id":"qf-mod47-morder-qualification","seed":202611167011,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":13}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=13 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(13, 2, 2, 1, 12, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (13, 2, 2, 1, 12, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0012","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64890ef94fbd5430f2fe1b8bebb4c0b0bd5a6200954b787a974ce938d82a1680","family_id":"qf-mod47-morder-qualification","seed":202611167012,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":14}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=14 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(14, 3, 3, 1, 6, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (14, 3, 3, 1, 6, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0013","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e6bb87d5fac57f3ac37a1270bd0847f85fc8a3568a044b8bee6b39b47b18e11","family_id":"qf-mod47-morder-qualification","seed":202611167013,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":15}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=15 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(15, 2, 2, 1, 8, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (15, 2, 2, 1, 8, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0014","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2870e3645883abe974d6c2c199d97bf27849509b22b40908f9ce3b6f4012847b","family_id":"qf-mod47-morder-qualification","seed":202611167014,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":16}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=16 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(16, 3, 3, 1, 8, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (16, 3, 3, 1, 8, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0015","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff9a876701a8a1dfab91fc3913b028a0ea17ab4b8416faf54a78531d07c34aef","family_id":"qf-mod47-morder-qualification","seed":202611167015,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":17}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(17, 2, 2, 1, 16, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (17, 2, 2, 1, 16, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0016","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3d43d9df4c589dd1457dc65c31ae21c093a8efc002980b9ac9cc2ac3903c96f","family_id":"qf-mod47-morder-qualification","seed":202611167016,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":18}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=18 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(18, 5, 5, 1, 6, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (18, 5, 5, 1, 6, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0017","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f61df1252b4436d2f62438605b2d785ac668327225d244d8971f2a9a6d0d44e8","family_id":"qf-mod47-morder-qualification","seed":202611167017,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":19}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=19 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(19, 2, 2, 1, 18, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (19, 2, 2, 1, 18, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0018","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bab3a1a04d9ab96011c4a73954290c4b7424556f13930dcaf7900e261449e14","family_id":"qf-mod47-morder-qualification","seed":202611167018,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":20}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(20, 3, 3, 1, 8, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (20, 3, 3, 1, 8, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0019","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce9b6bb51db8ca46a5ed2cee8a7fbc43c97686b7d2f3f60185d8f078475bab9","family_id":"qf-mod47-morder-qualification","seed":202611167019,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":21}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=21 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(21, 2, 2, 1, 12, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (21, 2, 2, 1, 12, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0020","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdb42adb4c68e3a4c3426b1ffa802c56ddd23bca2953bb63098fa85cdf4edb8c","family_id":"qf-mod47-morder-qualification","seed":202611167020,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":22}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=22 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(22, 3, 3, 1, 10, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (22, 3, 3, 1, 10, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0021","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e299b44bc470e61f9a54ac76f256a9997ac7feb430f525f93895329ccb5d1db","family_id":"qf-mod47-morder-qualification","seed":202611167021,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":23}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=23 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(23, 2, 2, 1, 22, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (23, 2, 2, 1, 22, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0022","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9877fc10baaa792c759fc4e8d7ef0f4f86e6353294e716fdf7c974ef19e11787","family_id":"qf-mod47-morder-qualification","seed":202611167022,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":24}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(24, 5, 5, 1, 8, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (24, 5, 5, 1, 8, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0023","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a6bc4050c3b90e1c48ad3d2c81861226f1b5f0ab4e08f930a672d3c8929fe24","family_id":"qf-mod47-morder-qualification","seed":202611167023,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":25}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(25, 2, 2, 1, 20, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (25, 2, 2, 1, 20, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0024","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9bf67de35ff96b26bb354e1f2ed76e3938f41f5108c6168162516e95761648a","family_id":"qf-mod47-morder-qualification","seed":202611167024,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":26}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=26 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(26, 3, 3, 1, 12, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (26, 3, 3, 1, 12, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0025","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f438582a2df40966ce8e677e8919eead16a020f969d74894e6a93a5c625030c","family_id":"qf-mod47-morder-qualification","seed":202611167025,"semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":27}"},"fields":{"title":"Normalize and Qualify a Unit before Defining Multiplicative Order","stem":"Use exact modulus n=27 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).","answer":{"kind":"expression","value":"(27, 2, 2, 1, 18, Yes, unit_mod_n_qualified)","explanation":"Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (27, 2, 2, 1, 18, Yes, unit_mod_n_qualified)."},"skills":["Normalize and Qualify a Unit before Defining Multiplicative Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0026","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bea284cfcdf3936b3ba55a12e47a49d8600a1f64624c4c1951e35afb6858e51","family_id":"qf-mod47-morder-power-first-return","seed":202611167026,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":3}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=3 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0027","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c41bb9eb8cd53b545530587c79e7e75ef6486b074b82153c151262f6b2525209","family_id":"qf-mod47-morder-power-first-return","seed":202611167027,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":4}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0028","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5c0474c9e731e1b52b41662ad45449e347f1d7701b851842abb2eb8dd12f429","family_id":"qf-mod47-morder-power-first-return","seed":202611167028,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":5}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=5 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 3], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 3], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0029","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b615a11e045170aee78b23acba50e928e9e739d3e457a4c893919f3b951bfdc1","family_id":"qf-mod47-morder-power-first-return","seed":202611167029,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":6}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=6 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0030","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f37d69cfcf1b9a61f02ffc2b853cac778e138d0ce7fcb11def20cafe7c7659","family_id":"qf-mod47-morder-power-first-return","seed":202611167030,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":7}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=7 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0031","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b6521bef4cb087d1eae1345beab83719b589d17c39965df4d27288439f5ca62","family_id":"qf-mod47-morder-power-first-return","seed":202611167031,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":8}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=8 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0032","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db9073bae0aa36f7f2287ea2d771a62d8f2869346b14858503b43e7cab2f4f0e","family_id":"qf-mod47-morder-power-first-return","seed":202611167032,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":9}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=9 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0033","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fc876d720f02ed26a0afe7e57ba17e91920bf111df24c764bfb68d79b427350","family_id":"qf-mod47-morder-power-first-return","seed":202611167033,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":10}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=10 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0034","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ecad26c2b363a2357eaf93790b1599efdcc43083dab25ce92f08e3bc2885dc3","family_id":"qf-mod47-morder-power-first-return","seed":202611167034,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":11}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=11 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], 10, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], 10, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0035","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1334c3f4674f94b58e6826208f325a36d3fc9c73ec8b4f7d2ed7337e3cb738fb","family_id":"qf-mod47-morder-power-first-return","seed":202611167035,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":12}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=12 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0036","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba3e480233537ef0d4a5cf62433ef69afb7156c24233c573fc93ba687554e170","family_id":"qf-mod47-morder-power-first-return","seed":202611167036,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":13}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=13 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], 12, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], 12, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0037","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d0f11de52157fbfa857529dd39d462c4832ba59914b69ab6dee660cbf143848","family_id":"qf-mod47-morder-power-first-return","seed":202611167037,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":14}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=14 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0038","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e23e729cfb5dee9b62504679144398f0e239339b3742bf1d7db8d3448de5a1d","family_id":"qf-mod47-morder-power-first-return","seed":202611167038,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":15}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=15 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0039","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e45eea669ead03f713ca73bdf793ca4403aa1fc27a7fa3c18cc477b43637b3b","family_id":"qf-mod47-morder-power-first-return","seed":202611167039,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":16}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=16 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 11], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 11], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0040","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a50ea7c913262e3281ec908dbf7eaddd18fbe7af9a8b0f2afa5ff5abfd84fc6","family_id":"qf-mod47-morder-power-first-return","seed":202611167040,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":17}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0041","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0e810f1fcfd3056a1091c7a4eb865173536c0ac512740a9cb10e09dd2bfeca0","family_id":"qf-mod47-morder-power-first-return","seed":202611167041,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":18}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=18 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0042","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e19bfa31173136d8d882e8f9585ac07a31c5b0aa2790da652f4c3f60356d500","family_id":"qf-mod47-morder-power-first-return","seed":202611167042,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":19}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=19 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], 18, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], 18, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0043","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff04d6c9c15688008198c14de4dd6ee876765773c439dddfd21f41a1a9fe9df7","family_id":"qf-mod47-morder-power-first-return","seed":202611167043,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":20}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0044","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d73d02608fb73822bf19f634aa5d49654cc5abdfa70358993cd15477b177ed16","family_id":"qf-mod47-morder-power-first-return","seed":202611167044,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":21}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=21 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0045","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6515f99c9163f3cf7f35cac40b5da4d52f5f249559138d059f323137c838ee65","family_id":"qf-mod47-morder-power-first-return","seed":202611167045,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":22}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=22 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0046","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abab1dcca3102af9b52df5c898927ca152a2a25982f79aafbd7cc224f6c0c652","family_id":"qf-mod47-morder-power-first-return","seed":202611167046,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":23}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=23 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], 11, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], 11, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0047","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74439d1c61edf88a665c9f233dd4d0c13729d74ba78f16dbc62fee54d4bc620c","family_id":"qf-mod47-morder-power-first-return","seed":202611167047,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":24}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0048","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8c6bcb2e49cfac6fb35970d3b15904cded33753db1df63049e5d5f5c907b859","family_id":"qf-mod47-morder-power-first-return","seed":202611167048,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":25}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], 20, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], 20, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0049","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58632e21fd454a8cb066dcce9bd308bdb9fc65e2064bf82240111409872dc1e0","family_id":"qf-mod47-morder-power-first-return","seed":202611167049,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":26}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=26 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0050","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81a36e34af28c602b94e3d077e0ee23e5088316c8d50158066b4def71197cb92","family_id":"qf-mod47-morder-power-first-return","seed":202611167050,"semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":27}"},"fields":{"title":"The First Positive Power Returning to One Defines the Order","stem":"Use exact modulus n=27 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], 18, Yes, Yes, order_is_the_first_positive_return)","explanation":"Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], 18, Yes, Yes, order_is_the_first_positive_return)."},"skills":["The First Positive Power Returning to One Defines the Order"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0051","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e09b98bdb92b1f3188ce9d2a14938bf364efeb3e056ae4b066491b59636c44f8","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167051,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":3}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=3 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(2, 2, Yes, [[1, 2], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 2, Yes, [[1, 2], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0052","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edd469e79783fd91c1e74454256feac180826b2788581ac885072050dbbe7586","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167052,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":4}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(2, 2, Yes, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 2, Yes, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0053","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad8ae64f7e19ef8849775370633e84a131832893b0b5939fd5957255c60281fd","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167053,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":5}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=5 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(4, 4, Yes, [[1, 2], [2, 4], [3, 3], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 4, Yes, [[1, 2], [2, 4], [3, 3], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0054","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"928198446e549c5dbee477f727fe0a092ba587d7bd89e4e5eb49cb519379bbce","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167054,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":6}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=6 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(2, 2, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 2, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0055","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb9068b2e163f20da44b5c8dbabf9a13bf4cef9de7d242fa89893d90ed372cc4","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167055,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":7}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=7 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(3, 6, Yes, [[1, 2], [2, 4], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (3, 6, Yes, [[1, 2], [2, 4], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0056","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c01d4e722c6e6f661e5315fec0399da74eb791d858724f3b8f4155c34af16481","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167056,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":8}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=8 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(2, 4, Yes, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 4, Yes, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0057","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c35e36cbffd8136c6a9fc5ab3b813eaa471c50e729c8406c550561522d222736","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167057,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":9}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=9 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(6, 6, Yes, [[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 6, Yes, [[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0058","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f8e854424fe95a74d662f78f7e8d63691f121b016d2e63530e01fb529c22e6d","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167058,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":10}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=10 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(4, 4, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 4, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0059","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1db3ef4bad6e392bcdbbf00da2ad026ea89cb298423922bcf8c92bf12fde8eed","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167059,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":11}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=11 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(10, 10, Yes, [[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (10, 10, Yes, [[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0060","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1399580fed128ab206fe80c7ae0e1868c7d6925ab62188ad0352fb904b2a4a31","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167060,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":12}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=12 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(2, 4, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 4, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0061","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec771b4ffa3102f626342101eab81df9be05aa2199d9da7d4e2bdcde135e8b2","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167061,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":13}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=13 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(12, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (12, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0062","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b9a6f89f0232e68f8a9d1429b697d0960bc9fb191e2cd76f36d6ed8e2943c13","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167062,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":14}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=14 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(6, 6, Yes, [[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 6, Yes, [[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0063","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6895051c17fe943e0a2188fd73a70d401714add0dec9fd8ebb98afc36d161a94","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167063,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":15}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=15 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(4, 8, Yes, [[1, 2], [2, 4], [3, 8], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 8, Yes, [[1, 2], [2, 4], [3, 8], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0064","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46a502a5f1e84324f23f210cf6fb27d8e249ce4a58a13d5a815eb26261a6768f","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167064,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":16}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=16 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(4, 8, Yes, [[1, 3], [2, 9], [3, 11], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 8, Yes, [[1, 3], [2, 9], [3, 11], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0065","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a714dfe2d179d2393433e2701490c424e60aca65d734d653a120f0f4572d146","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167065,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":17}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(8, 16, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (8, 16, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0066","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37701e108b70897c51acbd507d36ba5b34acf4124099248f068adea57a15ca5","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167066,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":18}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=18 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(6, 6, Yes, [[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 6, Yes, [[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0067","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a71f4264596b7274630dafaa4d441c11e2cc3bfddd838c0849ccb0ee4851df5b","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167067,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":19}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=19 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(18, 18, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (18, 18, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0068","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f3493cb9ae7cf37cc6ba52bd56f6f382632e8f3b6d637b839680f0ba262dae3","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167068,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":20}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(4, 8, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 8, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0069","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13598dca4cb3bb03c7927b7b4f4f7dac61259575d8aeb77a439660a1b049074f","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167069,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":21}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=21 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(6, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0070","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3b0ee7c4591499aaebcfbcf32a49f1851f2a18e7b4bd6f1a4cbf862f3a3a276","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167070,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":22}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=22 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(5, 10, Yes, [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (5, 10, Yes, [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0071","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3635ee558c2c3a347170afb9d1ad35951c7a0e832fba81fb78e230bf479e39e","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167071,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":23}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=23 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(11, 22, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (11, 22, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0072","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75f3b70c5a0fee4e78c3456bda90e3d0731788d7a553fe2a0b946d15dab684ba","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167072,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":24}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(2, 8, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 8, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0073","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ad340299a38ede0c99703a5b768a10b7367d9470e81abd84e3cddb4caef5670","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167073,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":25}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(20, 20, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (20, 20, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0074","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8d1bcc81efa031686f8a0823d9ca717336c8efd7e9a4e1bc93de4c3979b30e0","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167074,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":26}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=26 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(3, 12, Yes, [[1, 3], [2, 9], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (3, 12, Yes, [[1, 3], [2, 9], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0075","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21b8f7cbb77f16da3852e0b090a78e382b0b62bb8a349a75497ada2c67f76686","family_id":"qf-mod47-morder-phi-divisibility","seed":202611167075,"semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":27}"},"fields":{"title":"Order Divides Phi without Automatically Equaling Phi","stem":"Use exact modulus n=27 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).","answer":{"kind":"expression","value":"(18, 18, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation":"Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (18, 18, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)."},"skills":["Order Divides Phi without Automatically Equaling Phi"],"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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0076","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e33472bf80c6fc4b416eb9256a126fd1d1bde054734718690e3e9114f3d53e3","family_id":"qf-mod47-morder-boundary","seed":202611167076,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":8}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=8 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 8, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 8, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0077","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf10960f88ccccda151695de354a95a465ad8974c02b1ccba89675c0d0e474b","family_id":"qf-mod47-morder-boundary","seed":202611167077,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[3,1],\"mode\":\"invalid_modulus\",\"n\":1}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=1 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(invalid_modulus, 1, 3, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 3, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0078","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71e32a3149327b96b5110aabf1ac8ecb0dcfd043ecd8f7b42c97a92bc960b2fa","family_id":"qf-mod47-morder-boundary","seed":202611167078,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":5}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=5 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(order_one, 5, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 5, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0079","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bdb057221f9173fcd4d1754747cf10ddc746620fef98b4e711c40f1850aecc3","family_id":"qf-mod47-morder-boundary","seed":202611167079,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":8}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=8 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(negative_normalization, 8, -1, 7, 1, Yes, [[1, 7], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 8, -1, 7, 1, Yes, [[1, 7], [2, 1]], 2, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0080","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1ed426f53f5592a17d7a5c676c0ff66639c6e02e150268e4c4c45a8edf67921","family_id":"qf-mod47-morder-boundary","seed":202611167080,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":16}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=16 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 16, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 16, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0081","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad8d31fe22690511cbd2946155262df507ee02b05883a292cd84faaef00667ab","family_id":"qf-mod47-morder-boundary","seed":202611167081,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[7,1],\"mode\":\"invalid_modulus\",\"n\":1}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=1 and integer a=7. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(invalid_modulus, 1, 7, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 7, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0082","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7409980296d66c295126034d82797232354195523eb65d72fe62a41071f2867","family_id":"qf-mod47-morder-boundary","seed":202611167082,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":9}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=9 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(order_one, 9, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 9, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0083","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0fd442dd992d2fce776a752b3dcc7ea7d749b02a0826828d9d49e6970d1e676","family_id":"qf-mod47-morder-boundary","seed":202611167083,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":12}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=12 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(negative_normalization, 12, -1, 11, 1, Yes, [[1, 11], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 12, -1, 11, 1, Yes, [[1, 11], [2, 1]], 2, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0084","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90c6db07da4e198fb7e95442bce5b86b87348cb0c8e000ae072391b33e5a185a","family_id":"qf-mod47-morder-boundary","seed":202611167084,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":24}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=24 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 24, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 24, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0085","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da510e5545222b4066e4b1b233f2f3f548fd721eb1a35fa0cfe8e28733eee06d","family_id":"qf-mod47-morder-boundary","seed":202611167085,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[11,1],\"mode\":\"invalid_modulus\",\"n\":1}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=1 and integer a=11. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(invalid_modulus, 1, 11, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 11, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0086","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ab814cedfa75828b02af180c048f0e2eca3dc5d974c209965df49e4ef8c7922","family_id":"qf-mod47-morder-boundary","seed":202611167086,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":13}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=13 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(order_one, 13, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 13, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0087","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe7d80a27e7d7b7342a912fa76a28d7c31c7b410c024b9027ef50b1f703e2b57","family_id":"qf-mod47-morder-boundary","seed":202611167087,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":16}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=16 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(negative_normalization, 16, -1, 15, 1, Yes, [[1, 15], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 16, -1, 15, 1, Yes, [[1, 15], [2, 1]], 2, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0088","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e01b267f09e9a31c8dd8254e7eac4fd6118d2af5aa823b9587207975e1a42d4b","family_id":"qf-mod47-morder-boundary","seed":202611167088,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":32}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=32 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 32, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 32, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0089","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7514bffeb6c9f462f92fb0e119f84f74ff7cc44f4f79c55037e0314fa5cbbbf","family_id":"qf-mod47-morder-boundary","seed":202611167089,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[15,1],\"mode\":\"invalid_modulus\",\"n\":1}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=1 and integer a=15. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(invalid_modulus, 1, 15, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 15, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0090","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b352966cffa65163faf2e525987d522bcf0eb906f50ee55145435eeda4fc29e5","family_id":"qf-mod47-morder-boundary","seed":202611167090,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":17}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=17 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(order_one, 17, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 17, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0091","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d66d5f389e88bdd6f3d0e1c7a008b75c874b34ea909c9a283833c85249b8300","family_id":"qf-mod47-morder-boundary","seed":202611167091,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":20}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=20 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(negative_normalization, 20, -1, 19, 1, Yes, [[1, 19], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 20, -1, 19, 1, Yes, [[1, 19], [2, 1]], 2, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0092","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a43046ff55a7f25dad0f8c63e86521eedb7ea6151cdfbe765e362a525eff0343","family_id":"qf-mod47-morder-boundary","seed":202611167092,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":40}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=40 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 40, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 40, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0093","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c0bc1e188d5e1b971c8451ab585157bebcb164d0d660481d32dc9514e4762ab","family_id":"qf-mod47-morder-boundary","seed":202611167093,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[19,1],\"mode\":\"invalid_modulus\",\"n\":1}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=1 and integer a=19. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(invalid_modulus, 1, 19, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 19, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0094","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"369656824e3b7c848bd9435cccdc54683e08c976b8f1788f56aa70694d6568c9","family_id":"qf-mod47-morder-boundary","seed":202611167094,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":21}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=21 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(order_one, 21, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 21, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0095","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbdd34f137b8e6b730c16f00a38c35b1bfc189ecec99c09eaa9b685a4d02c4f1","family_id":"qf-mod47-morder-boundary","seed":202611167095,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":24}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=24 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(negative_normalization, 24, -1, 23, 1, Yes, [[1, 23], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 24, -1, 23, 1, Yes, [[1, 23], [2, 1]], 2, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0096","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"670ef2cda62eb6cc3a47ed40346e0a094563be8976f9f801348b7a7e4b38a76d","family_id":"qf-mod47-morder-boundary","seed":202611167096,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":48}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=48 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 48, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 48, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0097","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2fb485064b1bc3d9a3e99c97b1c265826f0de9e7fc43e6e68112bb9bb07a6b1","family_id":"qf-mod47-morder-boundary","seed":202611167097,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[23,1],\"mode\":\"invalid_modulus\",\"n\":1}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=1 and integer a=23. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(invalid_modulus, 1, 23, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 23, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0098","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb210f3312cafcf622b7a1b8ccd0a3098312e957c27302a093895aff38a4d84a","family_id":"qf-mod47-morder-boundary","seed":202611167098,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":25}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=25 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(order_one, 25, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 25, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0099","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"525ee1d1e8e76cb4a327826d854293a0b5d301d7380abc3af0a270733085d7e2","family_id":"qf-mod47-morder-boundary","seed":202611167099,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":28}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=28 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(negative_normalization, 28, -1, 27, 1, Yes, [[1, 27], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 28, -1, 27, 1, Yes, [[1, 27], [2, 1]], 2, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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-20261015-advance-20261116-r07-multiplicative-order-mod-n-0100","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b475db6fd137e312762c08b017f5f65365621d187b1597696fb7c8bb0f6bcc2c","family_id":"qf-mod47-morder-boundary","seed":202611167100,"semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":56}"},"fields":{"title":"Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues","stem":"Use exact modulus n=56 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).","answer":{"kind":"expression","value":"(noncoprime, 56, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)","explanation":"For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 56, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)."},"skills":["Undefined Nonunit/Modulus Cases Differ from Order One and Negative 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."}}