{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261018-r04-prime-number-0001","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32f566ad680b9b5246ca65b8e7d1a09592443b532a6bfde563d46397b88321b3","family_id":"qf-ext18-prime-divisor-classification","seed":202610184001,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":0}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=0. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(0, all-positive-naturals, No, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, all-positive-naturals, No, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0002","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b164f60dc515057f0dde34552c3252ab4cb2d49839f2dad8fb914fe8eb848e62","family_id":"qf-ext18-prime-divisor-classification","seed":202610184002,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":1}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=1. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(1, [1], No, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [1], No, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0003","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e452dad5be275b4ee9d5fdff7723e15398f814f02229fedce93a961cadb61c","family_id":"qf-ext18-prime-divisor-classification","seed":202610184003,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":2}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=2. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(2, [1, 2], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [1, 2], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0004","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"182a455d10bb693738cc1ad2527c6123e39b6361cb5d799314a2fe06fd100475","family_id":"qf-ext18-prime-divisor-classification","seed":202610184004,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":3}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=3. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(3, [1, 3], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [1, 3], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0005","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9712b0e4b0745553c85d5f0d6cf0bdc7b809253c0944dab6d3d22fdbe7fdfb9","family_id":"qf-ext18-prime-divisor-classification","seed":202610184005,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":4}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=4. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(4, [1, 2, 4], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2, 4], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0006","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"258dcb3598d133292c1560a7bb6e4ac675dc7f8961567b4175abe244f3e61046","family_id":"qf-ext18-prime-divisor-classification","seed":202610184006,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":5}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=5. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(5, [1, 5], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [1, 5], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0007","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee0430c8614ca85eca85e811a150bfa64627207a5b26d25b6c64252ee7ec741d","family_id":"qf-ext18-prime-divisor-classification","seed":202610184007,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":6}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=6. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(6, [1, 2, 3, 6], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [1, 2, 3, 6], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0008","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2256153364527586eeb5a1af6c4fad46c14baee461cf00076bdb33a3dbf46ae","family_id":"qf-ext18-prime-divisor-classification","seed":202610184008,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":7}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=7. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(7, [1, 7], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [1, 7], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0009","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dad5b790fdb79bbb27d3631bcd65a890b17219b1407c8380c1a6d600971d296d","family_id":"qf-ext18-prime-divisor-classification","seed":202610184009,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":8}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=8. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(8, [1, 2, 4, 8], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [1, 2, 4, 8], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0010","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6071efbefbac46aeac93804666a05e12c481812b2c0a1f6bfe69eeff1ae238f0","family_id":"qf-ext18-prime-divisor-classification","seed":202610184010,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":9}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=9. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(9, [1, 3, 9], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [1, 3, 9], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0011","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d34f9eec385ca03d86ffbc850d555eae12240e8807ada29a0a6f01cd0dc798b","family_id":"qf-ext18-prime-divisor-classification","seed":202610184011,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":10}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=10. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(10, [1, 2, 5, 10], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [1, 2, 5, 10], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0012","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed9525896fb4ee9143e621b27f4bcf5826b4f604d7b966c67f36a6dca80bef18","family_id":"qf-ext18-prime-divisor-classification","seed":202610184012,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":11}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=11. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(11, [1, 11], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, [1, 11], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0013","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b68fa51c85b046aa75e63d8c24360b99fc900b3b796cc44a241c94bf4e6a2a9a","family_id":"qf-ext18-prime-divisor-classification","seed":202610184013,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":12}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=12. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(12, [1, 2, 3, 4, 6, 12], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, [1, 2, 3, 4, 6, 12], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0014","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4111f1c8eda47ae5fd4432f02322284219445fe495a8c5828def8373aeadacc8","family_id":"qf-ext18-prime-divisor-classification","seed":202610184014,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":13}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=13. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(13, [1, 13], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, [1, 13], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0015","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abad882c6358c58dbc1521fa45f43cf84dbc1807a4f85f0dbb338ddcbf335e5d","family_id":"qf-ext18-prime-divisor-classification","seed":202610184015,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":14}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=14. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(14, [1, 2, 7, 14], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, [1, 2, 7, 14], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0016","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d87e2ddb52b24eaaebd600727689ae6c2e9a5ef7303616ff9caed238caefb725","family_id":"qf-ext18-prime-divisor-classification","seed":202610184016,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":15}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=15. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(15, [1, 3, 5, 15], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, [1, 3, 5, 15], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0017","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a000942482f8cbd92555b55130cf06c78fd0597c323f3540a8485e21915d512b","family_id":"qf-ext18-prime-divisor-classification","seed":202610184017,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":16}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=16. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(16, [1, 2, 4, 8, 16], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, [1, 2, 4, 8, 16], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0018","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da8c54e5a818900dea1a4930ded69df5cdd16a636bb8234037719542c454044c","family_id":"qf-ext18-prime-divisor-classification","seed":202610184018,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":17}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=17. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(17, [1, 17], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, [1, 17], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0019","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dd112ba29b63b6c29339a38199fafb1f01d41038aa634bdcfbc198aeae13f43","family_id":"qf-ext18-prime-divisor-classification","seed":202610184019,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":18}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=18. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(18, [1, 2, 3, 6, 9, 18], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, [1, 2, 3, 6, 9, 18], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0020","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc8ccccc7141ba88048011a4d40b2264149c177a5bc5241d9b09f8964335b5a7","family_id":"qf-ext18-prime-divisor-classification","seed":202610184020,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":19}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=19. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(19, [1, 19], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, [1, 19], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0021","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4070c0533635d0975f5ecba2632b56a53dde3608457bfffa8fda80076f7550a","family_id":"qf-ext18-prime-divisor-classification","seed":202610184021,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":20}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=20. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(20, [1, 2, 4, 5, 10, 20], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, [1, 2, 4, 5, 10, 20], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0022","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a340dba1a58f9ea2e3aa3275ead50d42307d5f9c8a6b2c10fcd785b6d40dbc7c","family_id":"qf-ext18-prime-divisor-classification","seed":202610184022,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":21}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=21. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(21, [1, 3, 7, 21], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, [1, 3, 7, 21], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0023","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef963160403719e94ebfc68d3053860eb7187068b2bc2b101e8fee47b0f447d","family_id":"qf-ext18-prime-divisor-classification","seed":202610184023,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":22}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=22. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(22, [1, 2, 11, 22], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, [1, 2, 11, 22], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0024","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98233cc02bf6996cb464f4e1d53b3a9efca484f0fc94e34fc60fefeb00da09d3","family_id":"qf-ext18-prime-divisor-classification","seed":202610184024,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":23}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=23. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(23, [1, 23], Yes, Yes)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, [1, 23], Yes, Yes)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0025","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28df82860bb486e69ae7c715065a5a5a9235349f913d579b55d8943d883c4aae","family_id":"qf-ext18-prime-divisor-classification","seed":202610184025,"semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":24}"},"fields":{"title":"Complete Positive Divisors and Prime Qualification","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=24. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).","answer":{"kind":"expression","value":"(24, [1, 2, 3, 4, 6, 8, 12, 24], Yes, No)","explanation":"Retain the lower-bound guard separately from the complete divisor inventory. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, [1, 2, 3, 4, 6, 8, 12, 24], Yes, No)."},"skills":["Complete Positive Divisors and Prime Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0026","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eff0b47db2cac8977d574d2465efc54620f1909a6d4bcf3bc2d0930132e9f98e","family_id":"qf-ext18-prime-smallest-factor","seed":202610184026,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":0}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=0. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(0, not-applicable, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, not-applicable, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0027","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4780e0701869f485ad2923dc1f8a089cfa13a420ff579ac2965bae3dc9fda970","family_id":"qf-ext18-prime-smallest-factor","seed":202610184027,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":1}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=1. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(1, not-applicable, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, not-applicable, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0028","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2205497373842f072fdb76527399885ebac0a18c18aa58e11cfb67581968dc","family_id":"qf-ext18-prime-smallest-factor","seed":202610184028,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":2}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=2. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(2, 2, Yes)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 2, Yes)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0029","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a2776204241784bce9f2e4e9c4188f96e383f0a72da2c8477287df26e896c29","family_id":"qf-ext18-prime-smallest-factor","seed":202610184029,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":33}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=33. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(33, 3, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, 3, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0030","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc0564d64589dfa6f0c42cfaba7f0de64796635e47ae0a2b8b6d85bd302a54b3","family_id":"qf-ext18-prime-smallest-factor","seed":202610184030,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":34}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=34. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(34, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0031","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d4d48ec9ec8a61eed377c547808d93165ccf76b1ce703607b6f9881d6d201d5","family_id":"qf-ext18-prime-smallest-factor","seed":202610184031,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":35}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=35. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(35, 5, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (35, 5, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0032","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da7596f970946078c382126e11b628ded56a06f56521601573f8eeefc9fc7438","family_id":"qf-ext18-prime-smallest-factor","seed":202610184032,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":36}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=36. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(36, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0033","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8baae1afa8f8590bea1844af9fb5917514408f858ab1079e30ac9c834ed1db2a","family_id":"qf-ext18-prime-smallest-factor","seed":202610184033,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":37}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=37. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(37, 37, Yes)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (37, 37, Yes)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0034","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"533ace903a11a1e5ef49d7c08750e070e4de14f5b82d2fd69482987415e7a196","family_id":"qf-ext18-prime-smallest-factor","seed":202610184034,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":38}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=38. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(38, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (38, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0035","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c33438ae5484f4f7f3bc9b82ddd44e0792f55efb06019858fa636d6b927d8388","family_id":"qf-ext18-prime-smallest-factor","seed":202610184035,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":39}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=39. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(39, 3, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (39, 3, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0036","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2504fa4c90966525806d615b23b7bae529d6e5abb38824af2439edf449624d2e","family_id":"qf-ext18-prime-smallest-factor","seed":202610184036,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":40}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=40. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(40, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (40, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0037","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ddf7138be316119357974e52b00a277f8b6b2bc7bd57d00a1fac6a086a83217","family_id":"qf-ext18-prime-smallest-factor","seed":202610184037,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":41}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=41. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(41, 41, Yes)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (41, 41, Yes)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0038","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5372659de31a668759baa7c0930fb0e02bb68c95b9548d41c13d6a119291cba8","family_id":"qf-ext18-prime-smallest-factor","seed":202610184038,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":42}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=42. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(42, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (42, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0039","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c4153d4482a38609451bbe0e8051aed8d7233be94b69b4d096680d4b228de81","family_id":"qf-ext18-prime-smallest-factor","seed":202610184039,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":43}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=43. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(43, 43, Yes)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (43, 43, Yes)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0040","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed5b8ffeec415dd6dc9bd7fd7412f091c9bb6a2416d2511ae110f901fc01bcac","family_id":"qf-ext18-prime-smallest-factor","seed":202610184040,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":44}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=44. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(44, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (44, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0041","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d694a5364bd28203da24e9863e5a8929c105ba471c828f622099058c0aff2c7d","family_id":"qf-ext18-prime-smallest-factor","seed":202610184041,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":45}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=45. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(45, 3, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (45, 3, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0042","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"babe123cae1a90532e85bdf80407f76c2fa93e6baa6d7651653cda5b039e09a2","family_id":"qf-ext18-prime-smallest-factor","seed":202610184042,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":46}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=46. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(46, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (46, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0043","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c633fe509465c77eedaf68d7d6c87d82a71e280886a4d01df657668768e2d534","family_id":"qf-ext18-prime-smallest-factor","seed":202610184043,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":47}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=47. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(47, 47, Yes)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (47, 47, Yes)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0044","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e7ae9e67ff947881cb5b40732c16eb0a31f20aee6976cd6bfcdae947450f15d","family_id":"qf-ext18-prime-smallest-factor","seed":202610184044,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":48}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=48. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(48, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (48, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0045","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a22bf9df5d9440226e8a4532212f7f4934cdd556c3d6f63c3040f59fffc55ea","family_id":"qf-ext18-prime-smallest-factor","seed":202610184045,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":49}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=49. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(49, 7, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (49, 7, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0046","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8866254dd80d61abc971ae164de1bedb8f02f29a398a0e7422b229b0306bcd88","family_id":"qf-ext18-prime-smallest-factor","seed":202610184046,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":50}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=50. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(50, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (50, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0047","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b465ab8cb0d448c85df0af816c390c4c1fbcb8251425c6f6e286e54e57c31091","family_id":"qf-ext18-prime-smallest-factor","seed":202610184047,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":51}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=51. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(51, 3, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (51, 3, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0048","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f850295e7bf2db4197ff1cbdb887d55665e9d09e590a46a3f6b77784a2562fe0","family_id":"qf-ext18-prime-smallest-factor","seed":202610184048,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":52}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=52. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(52, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (52, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0049","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d349a11505f006d421eff974f853f276a656e414f4f83b562ee6bb466615219","family_id":"qf-ext18-prime-smallest-factor","seed":202610184049,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":53}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=53. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(53, 53, Yes)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (53, 53, Yes)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0050","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b36750bcfd9e320eeb452828b1838f6d5cf694a15716e0eb6ec20043b8cf2b8c","family_id":"qf-ext18-prime-smallest-factor","seed":202610184050,"semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":54}"},"fields":{"title":"Least Divisor at Least Two and Primality","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=54. Give (n, least divisor d≥2 or not-applicable when n<2, prime?). For a prime input the least such divisor is n itself.","answer":{"kind":"expression","value":"(54, 2, No)","explanation":"Search through the complete square-root range before concluding that the input itself is the least factor. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (54, 2, No)."},"skills":["Least Divisor at Least Two and Primality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-prime-number-0051","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1468ef158f499bf3a8de36248e7743a3dc9c05b190c9a09f09cef7029947c70","family_id":"qf-ext18-prime-factor-witness","seed":202610184051,"semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":20}"},"fields":{"title":"Qualify a Nontrivial Composite-Factor Witness","stem":"Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. n=20, candidate d=2. Give (n, d, n≥2 and 2≤d