{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261018-r05-modular-congruence-0001","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d5aae40cc8ccab5a2f4de483e1e4fb75abfda0f204687e1ee3ad4f6d6c3138e","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185001,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":7,\"b\":12,\"n\":0}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=0, natural a=7, b=12. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(0, 7, 12, No, not-evaluated, not-evaluated)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 7, 12, No, not-evaluated, not-evaluated)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0002","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95cd373fab54afeef49d9d4c90d356800b35fa1423e5bc7e618d2f2e88ad99d9","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185002,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":10,\"b\":17,\"n\":1}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=1, natural a=10, b=17. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(1, 10, 17, Yes, [0, 0], Yes)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 10, 17, Yes, [0, 0], Yes)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0003","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d195a19b5366f2bf88d4ae67d28592e3ad67688ce63e03c04f3d587653550e22","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185003,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":13,\"b\":22,\"n\":2}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=2, natural a=13, b=22. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(2, 13, 22, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 13, 22, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0004","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f87ad022fd85fd97687a7bea0ead3ef5cf208931c9eb45d8206cb0105f7fc59d","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185004,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":16,\"b\":27,\"n\":3}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=3, natural a=16, b=27. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(3, 16, 27, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 16, 27, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0005","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7d80faff999f51863f98857200e7681e022793273faf7405f77c8a91b3ab190","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185005,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":19,\"b\":32,\"n\":4}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=4, natural a=19, b=32. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(4, 19, 32, Yes, [3, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 19, 32, Yes, [3, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0006","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12d5d60d446a6a52ca772cac7c03a928ae4420339c2860f4b6b34e8660f12f17","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185006,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":22,\"b\":37,\"n\":5}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=5, natural a=22, b=37. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(5, 22, 37, Yes, [2, 2], Yes)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 22, 37, Yes, [2, 2], Yes)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0007","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea6a7173a9cc74437a375678ab5db6d4c91349f87450b154ef7388e4a124ab8","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185007,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":25,\"b\":42,\"n\":0}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=0, natural a=25, b=42. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(0, 25, 42, No, not-evaluated, not-evaluated)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 25, 42, No, not-evaluated, not-evaluated)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0008","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"160f8fa359e66623afc4c7cda9ef111a781d958c1fcb470c0afd08e0a0dffb0d","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185008,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":28,\"b\":47,\"n\":1}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=1, natural a=28, b=47. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(1, 28, 47, Yes, [0, 0], Yes)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 28, 47, Yes, [0, 0], Yes)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0009","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0703ceece6492ae730e729ea5173f16fbdf7fa2550c314bb0e094ab159404f3f","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185009,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":31,\"b\":52,\"n\":2}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=2, natural a=31, b=52. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(2, 31, 52, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 31, 52, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0010","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c5d2b4f4872eddf88e1b0b34a7cad6c8071508fb011595b9c62436ff07476fb","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185010,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":34,\"b\":57,\"n\":3}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=3, natural a=34, b=57. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(3, 34, 57, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 34, 57, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0011","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d35491510f46a571d8ba718f2bd6170004e20170897ee761a56200ea14a2b29e","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185011,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":37,\"b\":62,\"n\":4}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=4, natural a=37, b=62. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(4, 37, 62, Yes, [1, 2], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 37, 62, Yes, [1, 2], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0012","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e48a6327566f1d9e60dcb17e4e8a82f6410f66b9ed5f22373a9a31fa5e769e5","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185012,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":40,\"b\":67,\"n\":5}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=5, natural a=40, b=67. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(5, 40, 67, Yes, [0, 2], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 40, 67, Yes, [0, 2], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0013","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9324abf89687101462c303c43169606a8af4d1dc9bd1aa99b7773c5a3519bb41","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185013,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":43,\"b\":72,\"n\":0}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=0, natural a=43, b=72. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(0, 43, 72, No, not-evaluated, not-evaluated)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 43, 72, No, not-evaluated, not-evaluated)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0014","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b010850c75238b393ccdecadc5e052d461414f0136b39f522cc4016eb9b7a1","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185014,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":46,\"b\":77,\"n\":1}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=1, natural a=46, b=77. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(1, 46, 77, Yes, [0, 0], Yes)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 46, 77, Yes, [0, 0], Yes)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0015","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8c19c0fbc1e2892d72fa6919a65cbea25c1377e62496136cba6c36ebbb27948","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185015,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":49,\"b\":82,\"n\":2}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=2, natural a=49, b=82. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(2, 49, 82, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 49, 82, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0016","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e59bd5bf971e35782928766aaf9d414aa643e3da31b409d8048c360b86afd0","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185016,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":52,\"b\":87,\"n\":3}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=3, natural a=52, b=87. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(3, 52, 87, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 52, 87, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0017","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fb300a61b50195d03742a8c8f01a6b8d0a2d31e141e8505417d383fa68c023c","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185017,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":55,\"b\":92,\"n\":4}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=4, natural a=55, b=92. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(4, 55, 92, Yes, [3, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 55, 92, Yes, [3, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0018","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"879713f565a5c1ee015abffcbc499086584f56f660f18146d6c4192c71f1bf26","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185018,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":58,\"b\":97,\"n\":5}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=5, natural a=58, b=97. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(5, 58, 97, Yes, [3, 2], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 58, 97, Yes, [3, 2], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0019","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaeb9e25a2ca384b7d14d089c0162181d093dad9b967744b65bd281987ed3714","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185019,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":61,\"b\":102,\"n\":0}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=0, natural a=61, b=102. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(0, 61, 102, No, not-evaluated, not-evaluated)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 61, 102, No, not-evaluated, not-evaluated)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0020","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77c81f3ddce928393d786731898b8bfb94766cf2ae1e2d9ed471f7d5a8be2741","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185020,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":64,\"b\":107,\"n\":1}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=1, natural a=64, b=107. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(1, 64, 107, Yes, [0, 0], Yes)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 64, 107, Yes, [0, 0], Yes)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0021","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0086b9bc6c2c3bd7db89d74cf3b0bbdc2d0ef80869978ef9b5a8a74b96a2fd8","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185021,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":67,\"b\":112,\"n\":2}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=2, natural a=67, b=112. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(2, 67, 112, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 67, 112, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0022","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b742fef0648246ffd95f4293ff0a95a361eaff3f1aa5225cc9be6d956eed42a","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185022,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":70,\"b\":117,\"n\":3}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=3, natural a=70, b=117. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(3, 70, 117, Yes, [1, 0], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 70, 117, Yes, [1, 0], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0023","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc9ad9f9941043a0a46e2923fab8baf8f2d4405655ebd84046aea96a2a3bb8f1","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185023,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":73,\"b\":122,\"n\":4}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=4, natural a=73, b=122. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(4, 73, 122, Yes, [1, 2], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 73, 122, Yes, [1, 2], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0024","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"affe4b07d5283d2845f54df3a3806789882d74ad4993b45a8dc99d8dec8d1769","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185024,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":76,\"b\":127,\"n\":5}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=5, natural a=76, b=127. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(5, 76, 127, Yes, [1, 2], No)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 76, 127, Yes, [1, 2], No)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0025","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"606baf7460a7586cc6060f1416d5ce9cdd13bbcd77b188748c427ec0ee8910c9","family_id":"qf-ext18-modeq-remainder-gate","seed":202610185025,"semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":79,\"b\":132,\"n\":0}"},"fields":{"title":"Positive-Modulus Guard before Natural Remainder Comparison","stem":"Fix modulus n=0, natural a=79, b=132. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.","answer":{"kind":"expression","value":"(0, 79, 132, No, not-evaluated, not-evaluated)","explanation":"Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 79, 132, No, not-evaluated, not-evaluated)."},"skills":["Positive-Modulus Guard before Natural Remainder Comparison"],"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-20261013-advance-20261018-r05-modular-congruence-0026","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"118b375df7069f38acd77af452b01ac87232cecafca8edf15946286b8be86363","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185026,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":1,\"bound\":25,\"n\":2}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=2, natural a=1, bound=25. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0027","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4241d3f8ac8896acd1bb8309c65c8e5085f6ce7649ae36ecb135977d120d1297","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185027,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":4,\"bound\":26,\"n\":3}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=3, natural a=4, bound=26. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(3, 4, 26, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 4, 26, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0028","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b831262f7a72e6b089a5918fed78ae184864bc17ccb5019903dd404b9b2d83c","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185028,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":7,\"bound\":27,\"n\":4}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=4, natural a=7, bound=27. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(4, 7, 27, 3, [3, 7, 11, 15, 19, 23, 27])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 7, 27, 3, [3, 7, 11, 15, 19, 23, 27])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0029","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"945a7c13275c470e203a35fa492dae4cb2d40da16c403f4b323ef42f0a15cd25","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185029,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":10,\"bound\":28,\"n\":5}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=5, natural a=10, bound=28. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(5, 10, 28, 0, [0, 5, 10, 15, 20, 25])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 10, 28, 0, [0, 5, 10, 15, 20, 25])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0030","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bda74d228347ebc2749eefcf7bab04e6c820fc92eac5d83a1d4e05cdc0f782d","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185030,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":13,\"bound\":29,\"n\":6}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=6, natural a=13, bound=29. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(6, 13, 29, 1, [1, 7, 13, 19, 25])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 13, 29, 1, [1, 7, 13, 19, 25])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0031","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1ae60298466762c8cd658186e180101fb062f3e04d0b9960818b8a60fb30866","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185031,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":16,\"bound\":30,\"n\":7}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=7, natural a=16, bound=30. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(7, 16, 30, 2, [2, 9, 16, 23, 30])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 16, 30, 2, [2, 9, 16, 23, 30])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0032","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e650c5acb7f89ab6cd875e094b5389d1e7fc4e382c060bfa183fdb0300e1f692","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185032,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":19,\"bound\":31,\"n\":8}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=8, natural a=19, bound=31. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(8, 19, 31, 3, [3, 11, 19, 27])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 19, 31, 3, [3, 11, 19, 27])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0033","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"626a0c2e345d26fffe877f4e422abeba79a18bd851428fc19a3a5a3c578cad36","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185033,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":22,\"bound\":32,\"n\":9}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=9, natural a=22, bound=32. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(9, 22, 32, 4, [4, 13, 22, 31])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 22, 32, 4, [4, 13, 22, 31])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0034","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53038e2d458e8b931f5d2c5c534f1860c9705a0f4d5ec8161b0cab99dc96ce0e","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185034,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":25,\"bound\":33,\"n\":10}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=10, natural a=25, bound=33. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(10, 25, 33, 5, [5, 15, 25])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 25, 33, 5, [5, 15, 25])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0035","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6aa6087ee9ab77a18be2563b792e61f9edb4c92975f4aabbd8d3b303d5bd8ba","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185035,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":28,\"bound\":34,\"n\":2}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=2, natural a=28, bound=34. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(2, 28, 34, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 28, 34, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0036","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3a195c8daaefe4a590c86e178443c257a95839aa1fae7aa8751eddecfa2942c","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185036,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":31,\"bound\":35,\"n\":3}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=3, natural a=31, bound=35. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0037","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d68471b2b209a11a73e147d0634dda1a3c353f1a764080ed470b2097ff1f78de","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185037,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":34,\"bound\":36,\"n\":4}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=4, natural a=34, bound=36. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(4, 34, 36, 2, [2, 6, 10, 14, 18, 22, 26, 30, 34])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 34, 36, 2, [2, 6, 10, 14, 18, 22, 26, 30, 34])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0038","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"174d307a3ec53c9871b771142a1107ebaf65dc3aa35022eedd22c51c4d083081","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185038,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":37,\"bound\":37,\"n\":5}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=5, natural a=37, bound=37. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(5, 37, 37, 2, [2, 7, 12, 17, 22, 27, 32, 37])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 37, 37, 2, [2, 7, 12, 17, 22, 27, 32, 37])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0039","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a30c2c31594cf65b7c328c63941a5c8516ce030928149fedd4ecccb097d3d9fa","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185039,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":40,\"bound\":38,\"n\":6}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=6, natural a=40, bound=38. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(6, 40, 38, 4, [4, 10, 16, 22, 28, 34])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 40, 38, 4, [4, 10, 16, 22, 28, 34])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0040","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"859cebc0f6f8c2fcc40f127d7fc8109ecd6b0c8b7bcec5f073615715d082c66f","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185040,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":43,\"bound\":39,\"n\":7}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=7, natural a=43, bound=39. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(7, 43, 39, 1, [1, 8, 15, 22, 29, 36])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 43, 39, 1, [1, 8, 15, 22, 29, 36])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0041","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a4e2e299681a7f5ce41a1ac7eec6aee2e9c50c8e87080dd89c799c327fa673d","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185041,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":46,\"bound\":40,\"n\":8}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=8, natural a=46, bound=40. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(8, 46, 40, 6, [6, 14, 22, 30, 38])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 46, 40, 6, [6, 14, 22, 30, 38])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0042","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce562fc3e9272a7dea7787fb7a12b4524265692ce53539540cd55c5a65f99878","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185042,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":49,\"bound\":41,\"n\":9}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=9, natural a=49, bound=41. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(9, 49, 41, 4, [4, 13, 22, 31, 40])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 49, 41, 4, [4, 13, 22, 31, 40])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0043","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7035386c8f7f1f4a5986e4e6e50eea1e2469f6328779e841353cb83f19fa6114","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185043,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":52,\"bound\":42,\"n\":10}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=10, natural a=52, bound=42. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(10, 52, 42, 2, [2, 12, 22, 32, 42])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 52, 42, 2, [2, 12, 22, 32, 42])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0044","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b21a6562f62503d6a52f94d6936554830376346553252a4e6c752d32f39b630","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185044,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":55,\"bound\":43,\"n\":2}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=2, natural a=55, bound=43. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(2, 55, 43, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 55, 43, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0045","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"833ed60660a3a522813a4d21bdf263b2d2fbae23e8d6efd5d89e66d56b676cfc","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185045,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":58,\"bound\":44,\"n\":3}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=3, natural a=58, bound=44. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(3, 58, 44, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 58, 44, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0046","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c162f40c125858a63537d9a1363c2e381e64b66f7072622480ffaa07d4dfec33","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185046,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":61,\"bound\":45,\"n\":4}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=4, natural a=61, bound=45. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(4, 61, 45, 1, [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 61, 45, 1, [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0047","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1477118cdaa56f262ee7b74d32f27d8e8a85d912a7036096d2f85934b3c1e47","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185047,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":64,\"bound\":46,\"n\":5}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=5, natural a=64, bound=46. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0048","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba90debd11dc9799b236d3875ba380848a837cbd4abccaceb307526b4981dd4","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185048,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":67,\"bound\":47,\"n\":6}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=6, natural a=67, bound=47. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(6, 67, 47, 1, [1, 7, 13, 19, 25, 31, 37, 43])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 67, 47, 1, [1, 7, 13, 19, 25, 31, 37, 43])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0049","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"605d7d3c0ba6632607954007232e9e642cd9f46e8ee2bd6ff62ccc479db5fb69","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185049,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":70,\"bound\":48,\"n\":7}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=7, natural a=70, bound=48. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(7, 70, 48, 0, [0, 7, 14, 21, 28, 35, 42])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 70, 48, 0, [0, 7, 14, 21, 28, 35, 42])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0050","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58896d69f8f3c276078d87ca3867e71818ace6b4cc84a203155b2b9029f7e51e","family_id":"qf-ext18-modeq-equivalence-class","seed":202610185050,"semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":73,\"bound\":49,\"n\":8}"},"fields":{"title":"A Natural Congruence Class in an Inclusive Window","stem":"Fix positive modulus n=8, natural a=73, bound=49. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).","answer":{"kind":"expression","value":"(8, 73, 49, 1, [1, 9, 17, 25, 33, 41, 49])","explanation":"Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 73, 49, 1, [1, 9, 17, 25, 33, 41, 49])."},"skills":["A Natural Congruence Class in an Inclusive Window"],"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-20261013-advance-20261018-r05-modular-congruence-0051","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d5802501fa90eea01a58ad6121015b17ac8097e9c3e996012977375706b1c17","family_id":"qf-ext18-modeq-add-preservation","seed":202610185051,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":1,\"b\":3,\"c\":3,\"d\":7,\"n\":2}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=2, natural inputs a=1, b=3, c=3, d=7. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(2, [1, 3], [3, 7], Yes, [0, 0], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [1, 3], [3, 7], Yes, [0, 0], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0052","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1786ab04214fa1556731373e0f5e3e6dae7fe566a6415a580b898280f3e89812","family_id":"qf-ext18-modeq-add-preservation","seed":202610185052,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":2,\"b\":8,\"c\":5,\"d\":14,\"n\":3}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=3, natural inputs a=2, b=8, c=5, d=14. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0053","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c5f5fa8ceffafea00dd7b726172bcb4b8cb21838170f16d342b7e33191713f","family_id":"qf-ext18-modeq-add-preservation","seed":202610185053,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":3,\"b\":15,\"c\":7,\"d\":11,\"n\":4}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=4, natural inputs a=3, b=15, c=7, d=11. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(4, [3, 15], [7, 11], Yes, [2, 2], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3, 15], [7, 11], Yes, [2, 2], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0054","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37ee68068a7442cab08de7fb6b3cb3ac3aad728f7e1f95656a4a28a9a972b98","family_id":"qf-ext18-modeq-add-preservation","seed":202610185054,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":4,\"b\":9,\"c\":9,\"d\":19,\"n\":5}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=5, natural inputs a=4, b=9, c=9, d=19. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(5, [4, 9], [9, 19], Yes, [3, 3], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [4, 9], [9, 19], Yes, [3, 3], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0055","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3a5019ea46ef4d25134394e9728c912cdd1993dbe4c89365d54a6fed29a3bc","family_id":"qf-ext18-modeq-add-preservation","seed":202610185055,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":5,\"b\":17,\"c\":11,\"d\":29,\"n\":6}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=6, natural inputs a=5, b=17, c=11, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(6, [5, 17], [11, 29], Yes, [4, 4], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [5, 17], [11, 29], Yes, [4, 4], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0056","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"826052a1c4eddc9259c990bdc32aabf3e821b01fe22160866b0ed6f76e7f3b79","family_id":"qf-ext18-modeq-add-preservation","seed":202610185056,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":6,\"b\":27,\"c\":13,\"d\":20,\"n\":7}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=7, natural inputs a=6, b=27, c=13, d=20. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(7, [6, 27], [13, 20], Yes, [5, 5], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [6, 27], [13, 20], Yes, [5, 5], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0057","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0934fda7104abb36ff5b0d1043405fd8e73a2c9a975a4d8d4c3fdbeaa46fe96","family_id":"qf-ext18-modeq-add-preservation","seed":202610185057,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":7,\"b\":15,\"c\":15,\"d\":31,\"n\":8}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=8, natural inputs a=7, b=15, c=15, d=31. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(8, [7, 15], [15, 31], Yes, [6, 6], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [7, 15], [15, 31], Yes, [6, 6], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0058","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1af6bf771090a739276689e6c487c19e7e76bda283781cecc27a8c4734bc096","family_id":"qf-ext18-modeq-add-preservation","seed":202610185058,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":8,\"b\":26,\"c\":17,\"d\":44,\"n\":9}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=9, natural inputs a=8, b=26, c=17, d=44. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(9, [8, 26], [17, 44], Yes, [7, 7], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [8, 26], [17, 44], Yes, [7, 7], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0059","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98128df63cebe1b083dd026fce00017890681a0928f8c1bfd818cd2e1bf2205","family_id":"qf-ext18-modeq-add-preservation","seed":202610185059,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":9,\"b\":39,\"c\":19,\"d\":29,\"n\":10}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=10, natural inputs a=9, b=39, c=19, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(10, [9, 39], [19, 29], Yes, [8, 8], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [9, 39], [19, 29], Yes, [8, 8], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0060","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ac71426b4fdf9dfc1694d1a81509c7b0386705038994535b58d16fd45513bb","family_id":"qf-ext18-modeq-add-preservation","seed":202610185060,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":10,\"b\":12,\"c\":21,\"d\":25,\"n\":2}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=2, natural inputs a=10, b=12, c=21, d=25. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(2, [10, 12], [21, 25], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [10, 12], [21, 25], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0061","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a21e39b1d032d923f7264e7b7f1336e4e3fa9e4ec4490a43048a3dbdd528f228","family_id":"qf-ext18-modeq-add-preservation","seed":202610185061,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":11,\"b\":17,\"c\":23,\"d\":32,\"n\":3}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=3, natural inputs a=11, b=17, c=23, d=32. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0062","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dccb7d729d5f6f61c0af0a86fb0e1c8c01394f85a109d6cd95053f58c7a16b45","family_id":"qf-ext18-modeq-add-preservation","seed":202610185062,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":12,\"b\":24,\"c\":25,\"d\":29,\"n\":4}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=4, natural inputs a=12, b=24, c=25, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(4, [12, 24], [25, 29], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [12, 24], [25, 29], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0063","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c2295598920447cafead3a4b26b8c694123a587edb5cc61818496bca7c48876","family_id":"qf-ext18-modeq-add-preservation","seed":202610185063,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":13,\"b\":18,\"c\":27,\"d\":37,\"n\":5}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=5, natural inputs a=13, b=18, c=27, d=37. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(5, [13, 18], [27, 37], Yes, [0, 0], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [13, 18], [27, 37], Yes, [0, 0], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0064","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a232afb83dd43a5520b0938921fd04867a22cce9c97075bfb286c4c0e14f42","family_id":"qf-ext18-modeq-add-preservation","seed":202610185064,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":14,\"b\":26,\"c\":29,\"d\":47,\"n\":6}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=6, natural inputs a=14, b=26, c=29, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(6, [14, 26], [29, 47], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [14, 26], [29, 47], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0065","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48fed075e572fe7140d0f6fe481285900dd406a0b6f847c73f3d45f0ee5d53fb","family_id":"qf-ext18-modeq-add-preservation","seed":202610185065,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":15,\"b\":36,\"c\":31,\"d\":38,\"n\":7}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=7, natural inputs a=15, b=36, c=31, d=38. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(7, [15, 36], [31, 38], Yes, [4, 4], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [15, 36], [31, 38], Yes, [4, 4], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0066","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1717497667b9b5665b850669420c4865ec79292241a765e9c8d23a32f7ccf030","family_id":"qf-ext18-modeq-add-preservation","seed":202610185066,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":16,\"b\":24,\"c\":33,\"d\":49,\"n\":8}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=8, natural inputs a=16, b=24, c=33, d=49. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(8, [16, 24], [33, 49], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [16, 24], [33, 49], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0067","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec81247d1e7702271fec837084660fc7cc603c8b981ea106adab860b231c83cd","family_id":"qf-ext18-modeq-add-preservation","seed":202610185067,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":17,\"b\":35,\"c\":35,\"d\":62,\"n\":9}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=9, natural inputs a=17, b=35, c=35, d=62. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(9, [17, 35], [35, 62], Yes, [7, 7], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [17, 35], [35, 62], Yes, [7, 7], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0068","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e1f111dbda4f49971f40a8f0d8904514263441b820a246a94d5b014cde94b32","family_id":"qf-ext18-modeq-add-preservation","seed":202610185068,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":18,\"b\":48,\"c\":37,\"d\":47,\"n\":10}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=10, natural inputs a=18, b=48, c=37, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(10, [18, 48], [37, 47], Yes, [5, 5], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [18, 48], [37, 47], Yes, [5, 5], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0069","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dfb34a4c390175b96746dd3f2a832ff0a446a8a1901ebfac4ced8ea1c8a0a0d","family_id":"qf-ext18-modeq-add-preservation","seed":202610185069,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":19,\"b\":21,\"c\":39,\"d\":43,\"n\":2}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=2, natural inputs a=19, b=21, c=39, d=43. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(2, [19, 21], [39, 43], Yes, [0, 0], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [19, 21], [39, 43], Yes, [0, 0], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0070","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77532b66bfe79108b5896841943619809887d5a387dfb1fa2aaf12fb5e2e184b","family_id":"qf-ext18-modeq-add-preservation","seed":202610185070,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":20,\"b\":26,\"c\":41,\"d\":50,\"n\":3}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=3, natural inputs a=20, b=26, c=41, d=50. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0071","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be4a71ec63d39d9169a00bb65aa206ba3e6754f80a1c6f1180835117e7125137","family_id":"qf-ext18-modeq-add-preservation","seed":202610185071,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":21,\"b\":33,\"c\":43,\"d\":47,\"n\":4}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(4, [21, 33], [43, 47], Yes, [0, 0], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [21, 33], [43, 47], Yes, [0, 0], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0072","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70c485a75dde7f68ee23ca216275d6324583d73619207701141d6b8e964a2d80","family_id":"qf-ext18-modeq-add-preservation","seed":202610185072,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":22,\"b\":27,\"c\":45,\"d\":55,\"n\":5}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=5, natural inputs a=22, b=27, c=45, d=55. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(5, [22, 27], [45, 55], Yes, [2, 2], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [22, 27], [45, 55], Yes, [2, 2], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0073","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70f7f5f198243b9cf5cc185d9058b1fc29410d561212b8225da1c8bb4a0445c4","family_id":"qf-ext18-modeq-add-preservation","seed":202610185073,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":23,\"b\":35,\"c\":47,\"d\":65,\"n\":6}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=6, natural inputs a=23, b=35, c=47, d=65. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(6, [23, 35], [47, 65], Yes, [4, 4], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [23, 35], [47, 65], Yes, [4, 4], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0074","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"823437da4e2fafef7114ef2c3b754d914fc1de55c56cc9cccfcf2c43c2ea988b","family_id":"qf-ext18-modeq-add-preservation","seed":202610185074,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":24,\"b\":45,\"c\":49,\"d\":56,\"n\":7}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=7, natural inputs a=24, b=45, c=49, d=56. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(7, [24, 45], [49, 56], Yes, [3, 3], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [24, 45], [49, 56], Yes, [3, 3], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0075","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d34a6a2f17149296a094e81133fe5ed3d83619b8ceebb81adc4f24f03dbcafe","family_id":"qf-ext18-modeq-add-preservation","seed":202610185075,"semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":25,\"b\":33,\"c\":51,\"d\":67,\"n\":8}"},"fields":{"title":"Additive Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=8, natural inputs a=25, b=33, c=51, d=67. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).","answer":{"kind":"expression","value":"(8, [25, 33], [51, 67], Yes, [4, 4], Yes, Yes)","explanation":"Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [25, 33], [51, 67], Yes, [4, 4], Yes, Yes)."},"skills":["Additive Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0076","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26ddb844c75c20d02d3f580636541cc069ca628799540fc536b73396dd531d49","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185076,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":1,\"b\":3,\"c\":3,\"d\":7,\"n\":2}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=2, natural inputs a=1, b=3, c=3, d=7. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(2, [1, 3], [3, 7], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [1, 3], [3, 7], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0077","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e7fbb2d3f8b7f57cfd7528704c1dd7bcb7ab52bd773ac6367746c585511aea","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185077,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":2,\"b\":8,\"c\":5,\"d\":14,\"n\":3}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=3, natural inputs a=2, b=8, c=5, d=14. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0078","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2216e88a87470f76988269d73b4d2ef3d5374fe87c2da2523110b7bbea3701e8","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185078,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":3,\"b\":15,\"c\":7,\"d\":11,\"n\":4}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=4, natural inputs a=3, b=15, c=7, d=11. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(4, [3, 15], [7, 11], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3, 15], [7, 11], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0079","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19520b327f2cc860db48c8412b9a235dc1e413f3dec0fea8569a5514d67b75ae","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185079,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":4,\"b\":9,\"c\":9,\"d\":19,\"n\":5}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=5, natural inputs a=4, b=9, c=9, d=19. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(5, [4, 9], [9, 19], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [4, 9], [9, 19], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0080","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c51bfbd364f33dee90997529c4e3cb5de56a79ee33ee37d87806745c9ffcfae6","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185080,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":5,\"b\":17,\"c\":11,\"d\":29,\"n\":6}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=6, natural inputs a=5, b=17, c=11, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(6, [5, 17], [11, 29], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [5, 17], [11, 29], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0081","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be2426dcfd59d6df5ad0a9d48eaf7066703b6a1d61a8545bf0b8df482748689b","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185081,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":6,\"b\":27,\"c\":13,\"d\":20,\"n\":7}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=7, natural inputs a=6, b=27, c=13, d=20. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(7, [6, 27], [13, 20], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [6, 27], [13, 20], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0082","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"add834f19480bf5a11e62dccf44f6551612f697fddfa9bfcefae5b829525554f","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185082,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":7,\"b\":15,\"c\":15,\"d\":31,\"n\":8}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=8, natural inputs a=7, b=15, c=15, d=31. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(8, [7, 15], [15, 31], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [7, 15], [15, 31], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0083","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b43680a1e18c07e86cc67b94a6a866adc5ce1cf213e04c559a73af3b31ac1be5","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185083,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":8,\"b\":26,\"c\":17,\"d\":44,\"n\":9}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=9, natural inputs a=8, b=26, c=17, d=44. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(9, [8, 26], [17, 44], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [8, 26], [17, 44], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0084","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ab325d23b3e829006aada989d263f2ebc29cf857fe60d08fd6f6512afb6cca7","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185084,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":9,\"b\":39,\"c\":19,\"d\":29,\"n\":10}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=10, natural inputs a=9, b=39, c=19, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(10, [9, 39], [19, 29], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [9, 39], [19, 29], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0085","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9922d60ac45bacf93d382d990e76fc174417d0ae32b6d6af11cbc0ee07d883e","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185085,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":10,\"b\":12,\"c\":21,\"d\":25,\"n\":2}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=2, natural inputs a=10, b=12, c=21, d=25. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(2, [10, 12], [21, 25], Yes, [0, 0], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [10, 12], [21, 25], Yes, [0, 0], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0086","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37bf8289930eb915be03f72b6661a5b75b6215ce043939406c037c04e46acb69","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185086,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":11,\"b\":17,\"c\":23,\"d\":32,\"n\":3}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=3, natural inputs a=11, b=17, c=23, d=32. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0087","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8740e8ed6c3fbe43ce4665e37836667a3abc979fc957e951daa5cee2d6b61db","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185087,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":12,\"b\":24,\"c\":25,\"d\":29,\"n\":4}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=4, natural inputs a=12, b=24, c=25, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(4, [12, 24], [25, 29], Yes, [0, 0], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [12, 24], [25, 29], Yes, [0, 0], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0088","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c4d921e93c2331b681bd56fa7b847db94e431397cddeebc55346c684e0f821e","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185088,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":13,\"b\":18,\"c\":27,\"d\":37,\"n\":5}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=5, natural inputs a=13, b=18, c=27, d=37. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(5, [13, 18], [27, 37], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [13, 18], [27, 37], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0089","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a772a2a960aa1361ec8848b8f6b03e4bc88154eeed691dc02eee9a673f97dbe","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185089,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":14,\"b\":26,\"c\":29,\"d\":47,\"n\":6}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=6, natural inputs a=14, b=26, c=29, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(6, [14, 26], [29, 47], Yes, [4, 4], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [14, 26], [29, 47], Yes, [4, 4], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0090","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19543ed136b630763ac8af13110ffa7f8d721c1cb6fccef01bc5bec9809012e1","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185090,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":15,\"b\":36,\"c\":31,\"d\":38,\"n\":7}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=7, natural inputs a=15, b=36, c=31, d=38. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(7, [15, 36], [31, 38], Yes, [3, 3], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [15, 36], [31, 38], Yes, [3, 3], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0091","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb9d1202b01bb37d7d3312eaedb399e156eede54bb25133cc5393aed1182eb74","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185091,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":16,\"b\":24,\"c\":33,\"d\":49,\"n\":8}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=8, natural inputs a=16, b=24, c=33, d=49. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(8, [16, 24], [33, 49], Yes, [0, 0], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [16, 24], [33, 49], Yes, [0, 0], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0092","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2842ae496d58deed39d5c5dbb2d3a8545bd659fbb4350ec26a0d1002971fc6c9","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185092,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":17,\"b\":35,\"c\":35,\"d\":62,\"n\":9}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=9, natural inputs a=17, b=35, c=35, d=62. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(9, [17, 35], [35, 62], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [17, 35], [35, 62], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0093","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"150fe3988e9470b54d959a7bfeecf4171a56f4b4fb5433f8446b902d091ca12c","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185093,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":18,\"b\":48,\"c\":37,\"d\":47,\"n\":10}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=10, natural inputs a=18, b=48, c=37, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(10, [18, 48], [37, 47], Yes, [6, 6], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [18, 48], [37, 47], Yes, [6, 6], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0094","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"895ac5b1955b40acd9db0bff5879241c17bd65cd3b0ad3a91794150b395204f1","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185094,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":19,\"b\":21,\"c\":39,\"d\":43,\"n\":2}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=2, natural inputs a=19, b=21, c=39, d=43. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(2, [19, 21], [39, 43], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [19, 21], [39, 43], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0095","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39eeb430dbba8b009dcccb26a473b0358396e71ec32949796dfc8acb510b2b75","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185095,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":20,\"b\":26,\"c\":41,\"d\":50,\"n\":3}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=3, natural inputs a=20, b=26, c=41, d=50. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0096","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be1f5b84eac7bea6439d4a2ead7c78d5466d53258de9d245372659ea2bf1da23","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185096,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":21,\"b\":33,\"c\":43,\"d\":47,\"n\":4}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(4, [21, 33], [43, 47], Yes, [3, 3], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [21, 33], [43, 47], Yes, [3, 3], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0097","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27973f7ec5c78fc9305ee16a48fab4183451510302a13aab8541f71ffa1c2e8","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185097,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":22,\"b\":27,\"c\":45,\"d\":55,\"n\":5}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=5, natural inputs a=22, b=27, c=45, d=55. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(5, [22, 27], [45, 55], Yes, [0, 0], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [22, 27], [45, 55], Yes, [0, 0], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0098","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cd927464967fdd6255508b786e972bd609f20988cd2e780bcb390d0ad64a259","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185098,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":23,\"b\":35,\"c\":47,\"d\":65,\"n\":6}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=6, natural inputs a=23, b=35, c=47, d=65. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(6, [23, 35], [47, 65], Yes, [1, 1], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [23, 35], [47, 65], Yes, [1, 1], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0099","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a070aaadccbcf8571b2dfd632e30408666e84d301f5ca82cc5bc867fc3b6385","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185099,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":24,\"b\":45,\"c\":49,\"d\":56,\"n\":7}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=7, natural inputs a=24, b=45, c=49, d=56. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(7, [24, 45], [49, 56], Yes, [0, 0], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [24, 45], [49, 56], Yes, [0, 0], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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-20261013-advance-20261018-r05-modular-congruence-0100","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f69b396f6436aa848a2c85606ee340c810ab92a4114f7ed275bd596e5e4b57a5","family_id":"qf-ext18-modeq-mul-preservation","seed":202610185100,"semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":25,\"b\":33,\"c\":51,\"d\":67,\"n\":8}"},"fields":{"title":"Multiplicative Congruence Values, Premises, and Material Implication","stem":"Use fixed positive modulus n=8, natural inputs a=25, b=33, c=51, d=67. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.","answer":{"kind":"expression","value":"(8, [25, 33], [51, 67], Yes, [3, 3], Yes, Yes)","explanation":"Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [25, 33], [51, 67], Yes, [3, 3], Yes, Yes)."},"skills":["Multiplicative Congruence Values, Premises, and Material Implication"],"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."}}