{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261018-r02-natural-divisibility-0001","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"426736e3a599331b89aa72c2e9327a6f58c3fec8d03fe376f2bbe118a621ddee","family_id":"qf-ext18-divisibility-witness","seed":202610182001,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":0,\"b\":0}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, b=0. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(0, 0, Yes, 0)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, Yes, 0)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0002","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5e0b2e6f1a72e3dde47520ec4c81d31aaa0b780616fb0a25c3ae375797637f","family_id":"qf-ext18-divisibility-witness","seed":202610182002,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":0,\"b\":7}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, b=7. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(0, 7, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 7, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0003","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95ee836f00cffff58983eab82272f4e528f238b343c91b8e4cba2a89a40f5551","family_id":"qf-ext18-divisibility-witness","seed":202610182003,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":2,\"b\":7}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, b=7. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(2, 7, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 7, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0004","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ebbfd111de201827384dd33683f563ea68e047727f8969b3c5013e0cdc48869","family_id":"qf-ext18-divisibility-witness","seed":202610182004,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":3,\"b\":12}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, b=12. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(3, 12, Yes, 4)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 12, Yes, 4)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0005","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69ab9da41a482815267992400b2fad497c31d8193f7b2f959cda22dea5e4042d","family_id":"qf-ext18-divisibility-witness","seed":202610182005,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":4,\"b\":21}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, b=21. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(4, 21, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 21, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0006","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b87a32a9e1d39275f9bc298e96e01f47a8845cf42b4f9c873104a549ceb0d093","family_id":"qf-ext18-divisibility-witness","seed":202610182006,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":5,\"b\":30}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, b=30. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(5, 30, Yes, 6)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 30, Yes, 6)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0007","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de9bebeb21a71ba8b410a0e6685b123227473dbb15c53ca2ce6038185dec40d5","family_id":"qf-ext18-divisibility-witness","seed":202610182007,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":6,\"b\":43}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=6, b=43. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(6, 43, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 43, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0008","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6f829a879ee035401ea2be6bcba0c3e00b84473197dac6d742bfa4f8a103e72","family_id":"qf-ext18-divisibility-witness","seed":202610182008,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":7,\"b\":56}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=7, b=56. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(7, 56, Yes, 8)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 56, Yes, 8)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0009","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96e04558df9993013e09154d95fdfcfbfe4c3e3c0531dfcb146ce0efd8f53cf","family_id":"qf-ext18-divisibility-witness","seed":202610182009,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":8,\"b\":73}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=8, b=73. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(8, 73, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 73, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0010","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fa72b88eba3c4240171499687186f8517ea1e78b9dc28d59c8b8171cb9a8e1c","family_id":"qf-ext18-divisibility-witness","seed":202610182010,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":9,\"b\":90}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=9, b=90. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(9, 90, Yes, 10)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 90, Yes, 10)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0011","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71584772562bbf2c5455bbbd33ff32557b43addcbb58abfb2b976e9a548c4540","family_id":"qf-ext18-divisibility-witness","seed":202610182011,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":10,\"b\":111}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=10, b=111. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(10, 111, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 111, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0012","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d66e2e6552fe5697bfcd4e56a3a9761e1e4e88cb0aaea290e49d700a84a70c7","family_id":"qf-ext18-divisibility-witness","seed":202610182012,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":11,\"b\":132}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=11, b=132. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(11, 132, Yes, 12)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, 132, Yes, 12)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0013","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e82cf1d8e74ecadcfb31f25b7324ee2f314185226c76879187740e6bd75ac84a","family_id":"qf-ext18-divisibility-witness","seed":202610182013,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":12,\"b\":157}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=12, b=157. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(12, 157, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 157, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0014","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74f25cf9d73c51d1ee9a1b695f23f050e54e13a6d05ec97062242e99787cadc4","family_id":"qf-ext18-divisibility-witness","seed":202610182014,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":13,\"b\":182}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=13, b=182. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(13, 182, Yes, 14)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 182, Yes, 14)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0015","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810e710f1d8a74a141e89350dd73a6370cdb67915e63632880be01acd8fd7c8c","family_id":"qf-ext18-divisibility-witness","seed":202610182015,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":14,\"b\":211}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=14, b=211. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(14, 211, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 211, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0016","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a299e0b3b192303fadbc66f5e5bd7ca5e0769c39cb059b65c2629d6f42b9792","family_id":"qf-ext18-divisibility-witness","seed":202610182016,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":15,\"b\":240}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=15, b=240. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(15, 240, Yes, 16)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 240, Yes, 16)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0017","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ff52a74cb3a4705db0b5bba9443be7e1546c339516da178d96f07b948440376","family_id":"qf-ext18-divisibility-witness","seed":202610182017,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":16,\"b\":273}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=16, b=273. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(16, 273, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 273, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0018","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55667b1a4e1f8ccffaa101b72e34b6097ae0771ef5313216a492fd80e4d4281d","family_id":"qf-ext18-divisibility-witness","seed":202610182018,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":17,\"b\":306}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=17, b=306. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(17, 306, Yes, 18)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 306, Yes, 18)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0019","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5433357a93458015ad661f5b82a84bbe39e253d93d465a34a88ea56c42796118","family_id":"qf-ext18-divisibility-witness","seed":202610182019,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":18,\"b\":343}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=18, b=343. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(18, 343, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 343, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0020","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cb12d186a80d4ddd1d1ae4e52cb6cb7e0cf940cfd923be5c2c9d64eaf757115","family_id":"qf-ext18-divisibility-witness","seed":202610182020,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":19,\"b\":380}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=19, b=380. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(19, 380, Yes, 20)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 380, Yes, 20)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0021","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"108944325807dcee5ee4f7e1662d3c13226b86c9052d3e29d1600767e8062609","family_id":"qf-ext18-divisibility-witness","seed":202610182021,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":20,\"b\":421}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=20, b=421. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(20, 421, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 421, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0022","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e725eb98070739e63a284019f1c61f846beb0a3a056aff9808670abe0da579a","family_id":"qf-ext18-divisibility-witness","seed":202610182022,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":21,\"b\":462}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=21, b=462. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(21, 462, Yes, 22)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 462, Yes, 22)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0023","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9bc13be63177d4110cce5521a83166c76bab9b19d608ccd9822301cf0a63258","family_id":"qf-ext18-divisibility-witness","seed":202610182023,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":22,\"b\":507}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=22, b=507. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(22, 507, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 507, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0024","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96b5fe9635592e3c4ab3370bcd7ba4ba9ca02ad02e37db660c3692bfb134ddb4","family_id":"qf-ext18-divisibility-witness","seed":202610182024,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":23,\"b\":552}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=23, b=552. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(23, 552, Yes, 24)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 552, Yes, 24)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0025","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1a6993653826be1f29c64c0cd376714fc97bcb75cdfa831e636471e766aedd","family_id":"qf-ext18-divisibility-witness","seed":202610182025,"semantic_template":"qf-ext18-divisibility-witness/{\"a\":24,\"b\":601}"},"fields":{"title":"Natural Divisibility and an Explicit Multiplier Witness","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=24, b=601. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.","answer":{"kind":"expression","value":"(24, 601, No, does-not-exist)","explanation":"Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 601, No, does-not-exist)."},"skills":["Natural Divisibility and an Explicit Multiplier Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0026","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ac5eb8a1d773434df82883bc195b25e920be10716de9865c2f9488e68b65ea0","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182026,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":0}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=0. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(0, all-positive-naturals, infinite)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, all-positive-naturals, infinite)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0027","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4afa8fc67bcc249cf6ed91dbc23483c1448bd1a0c7b3def7971544269d55a05c","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182027,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":11}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=11. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(11, [1, 11], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, [1, 11], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0028","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a2d16f52137ccd65bf6a6ab065c9dc3a2bf593293876df1189d3110637948b8","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182028,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":12}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=12. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(12, [1, 2, 3, 4, 6, 12], 6)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. 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], 6)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0029","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25bb635390428fc378dcdfeaa7af2fe96250dbd551b5ef469503d2a6c253883b","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182029,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":13}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=13. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(13, [1, 13], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, [1, 13], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0030","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e78a38295ac06f298dbdbdf3f50636fec9aaf086767b8a240decfc61ce81369","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182030,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":14}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=14. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(14, [1, 2, 7, 14], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, [1, 2, 7, 14], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0031","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"780eb58931acb54512340328c4cfceb4a5a4405d1507dea06db21bed308edaad","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182031,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":15}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=15. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(15, [1, 3, 5, 15], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, [1, 3, 5, 15], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0032","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62a13ca4797186aa1f0c63936fbb0ad5cff1a04265b6ec4dd0a85a784915acc0","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182032,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":16}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=16. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(16, [1, 2, 4, 8, 16], 5)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, [1, 2, 4, 8, 16], 5)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0033","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"207e9b35bad20b10c2128b0fec7c6eb7d60f9636adbd4ad52c6276b341759475","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182033,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":17}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=17. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(17, [1, 17], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, [1, 17], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0034","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb7bdfc405bd00a4042030d05fed28b5a6d29a5e3eda2cdf548fbbe227ce0af8","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182034,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":18}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=18. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(18, [1, 2, 3, 6, 9, 18], 6)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. 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], 6)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0035","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60e3524a7712a8f9a71f359097bea911e828cda8abf094995ef7e6960a720a0a","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182035,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":19}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=19. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(19, [1, 19], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, [1, 19], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0036","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a2db0b7f213e23af918484c190ef03b77c583e357c0835b069c2c62b0e9200","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182036,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":20}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=20. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(20, [1, 2, 4, 5, 10, 20], 6)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. 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], 6)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0037","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"568a42646a6774747f2f20e8e1eb726e84f8e7689806fc6b14fa95da8ee10c84","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182037,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":21}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=21. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(21, [1, 3, 7, 21], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, [1, 3, 7, 21], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0038","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ec51a0e30e080edab93b601ecf8c24473cf1fa61051976b452cfeb104b95330","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182038,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":22}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=22. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(22, [1, 2, 11, 22], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, [1, 2, 11, 22], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0039","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0b9258e5834b1fa8acd2c773d63b80f7ca9f3ca8ee45ef5b332a01c6d74874d","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182039,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":23}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=23. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(23, [1, 23], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, [1, 23], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0040","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02c83c03c3053e34bb7258445600584e4b7be03ab9a99207fd4c7ae0c6eaeb0","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182040,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":24}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=24. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(24, [1, 2, 3, 4, 6, 8, 12, 24], 8)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. 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], 8)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0041","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"477d7deddd3afeb877e31b0143eba78608154567f8480f464b3a59bd6b940f1e","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182041,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":25}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=25. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(25, [1, 5, 25], 3)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (25, [1, 5, 25], 3)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0042","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7a1821c7f3a2be27be8e2af71513151f2c105709176b857ba7cd6303d60e9d","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182042,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":26}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=26. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(26, [1, 2, 13, 26], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (26, [1, 2, 13, 26], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0043","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51dcb3c75cf38398e1957c72d64353019b3ffb318dffcf78c73586fb358b0b40","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182043,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":27}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=27. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(27, [1, 3, 9, 27], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (27, [1, 3, 9, 27], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0044","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c73405ed5272daac5be16029702a09323d8eddf03c7758173adc92f9a4dfcc77","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182044,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":28}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=28. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(28, [1, 2, 4, 7, 14, 28], 6)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (28, [1, 2, 4, 7, 14, 28], 6)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0045","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5343ae49ace6be02324f5ca998f72428ce95e63bd9105ad06e0aacb69385a273","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182045,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":29}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=29. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(29, [1, 29], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (29, [1, 29], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0046","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d61c0446c07279e818853d34cef67aaeb1ece20f6db90830085da19999e4b5c","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182046,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":30}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=30. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, [1, 2, 3, 5, 6, 10, 15, 30], 8)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0047","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14379ea2e53246c0471d0c3c9897e2bb2217bda7862819103aa51905a2bc09ce","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182047,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":31}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=31. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(31, [1, 31], 2)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (31, [1, 31], 2)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0048","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb8642d3e40687bb2d66715872c08eb03ea24971c0c3d7507e28a05aa7dfe1b7","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182048,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":32}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=32. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(32, [1, 2, 4, 8, 16, 32], 6)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (32, [1, 2, 4, 8, 16, 32], 6)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0049","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80060cef710413d73b2ec5e5515c69a8bf43d7d02ff1476599f662825eddd4e3","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182049,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":33}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=33. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(33, [1, 3, 11, 33], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, [1, 3, 11, 33], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0050","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fb9d6e3459bc9d02b690d72120847762e3d11ca03ba2be7dbbbe5956bf11b17","family_id":"qf-ext18-divisibility-divisor-set","seed":202610182050,"semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":34}"},"fields":{"title":"Complete Positive Divisor Inventory Including the Zero Case","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=34. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.","answer":{"kind":"expression","value":"(34, [1, 2, 17, 34], 4)","explanation":"For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, [1, 2, 17, 34], 4)."},"skills":["Complete Positive Divisor Inventory Including the Zero Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0051","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3cb11c4a456442028aed1ef819fa02c02adeab953e3b281f8062079bec2958e","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182051,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":0,\"bound\":30}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, b=0, bound=30. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(0, 0, 30, [0])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 30, [0])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0052","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7c196bf5bdb8073baf080a87a11cbf996b74a7af70993c6b6f46a3eca5e3b3c","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182052,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":2,\"bound\":31}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, b=2, bound=31. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(1, 2, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0053","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a4372f2f89a8a655ff5968f71255bfeae3af4fee82d54c8b60881d439885149","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182053,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":4,\"bound\":32}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, b=4, bound=32. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(2, 4, 32, [0, 4, 8, 12, 16, 20, 24, 28, 32])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 4, 32, [0, 4, 8, 12, 16, 20, 24, 28, 32])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0054","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbc2c998b5e6ed99d84bf1b48152bfa685b2e357f68e9fb000da72f79fca4740","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182054,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":6,\"bound\":33}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, b=6, bound=33. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(3, 6, 33, [0, 6, 12, 18, 24, 30])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 6, 33, [0, 6, 12, 18, 24, 30])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0055","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76bddf73675af071bed5446aab920093fe19d9f8ef6deafd71c0c5bbe4b88793","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182055,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":4,\"b\":8,\"bound\":34}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, b=8, bound=34. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(4, 8, 34, [0, 8, 16, 24, 32])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 8, 34, [0, 8, 16, 24, 32])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0056","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a93f06089d1c25e3ab90bf981e3e87db156f6a215f24edb7cba06d7ee905b04c","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182056,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":5,\"b\":1,\"bound\":35}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, b=1, bound=35. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(5, 1, 35, [0, 5, 10, 15, 20, 25, 30, 35])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 1, 35, [0, 5, 10, 15, 20, 25, 30, 35])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0057","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5e6d8cd5fc14c7d59e9a5457f0c308e76b4f0d2bbc1e3cfb6f0b9de9af004f","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182057,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":3,\"bound\":36}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=6, b=3, bound=36. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(6, 3, 36, [0, 6, 12, 18, 24, 30, 36])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 3, 36, [0, 6, 12, 18, 24, 30, 36])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0058","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0fcef3e77652342db5d8dd24580864b749110b5eadb084182e53fdf7f8986d3","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182058,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":5,\"bound\":37}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, b=5, bound=37. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(0, 5, 37, [0])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 5, 37, [0])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0059","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6e102a477b3e1860210bd78c92ce2462fd67a9a6df0ce96b35f9f1aab9d15c1","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182059,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":7,\"bound\":38}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, b=7, bound=38. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(1, 7, 38, [0, 7, 14, 21, 28, 35])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 7, 38, [0, 7, 14, 21, 28, 35])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0060","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8596438bd7a8abdb20f9bcce304615632b8814038683dc0116dcf44016e78ad","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182060,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":0,\"bound\":39}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, b=0, bound=39. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(2, 0, 39, [0])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 0, 39, [0])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0061","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3273537dc27eaa68a6e5f360df5af37187cbe30e5c6c02f45cc8654701dfb6d","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182061,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":2,\"bound\":40}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, b=2, bound=40. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(3, 2, 40, [0, 6, 12, 18, 24, 30, 36])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 2, 40, [0, 6, 12, 18, 24, 30, 36])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0062","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b695d2e67d8b827f5bc5d9e88c445fd91fba961b5231f0f03705b736199068f1","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182062,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":4,\"b\":4,\"bound\":41}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, b=4, bound=41. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(4, 4, 41, [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, 41, [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0063","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1243d4132a8827aae3ca74ec569beeb6d55adacefbdbde79b239932cb84193a","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182063,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":5,\"b\":6,\"bound\":42}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, b=6, bound=42. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(5, 6, 42, [0, 30])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 6, 42, [0, 30])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0064","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a6cf50ab059b61170bb0f8b81dd1cc52d28d19ce6d24a458e2b17e0a0bd5748","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182064,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":8,\"bound\":43}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=6, b=8, bound=43. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(6, 8, 43, [0, 24])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 8, 43, [0, 24])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0065","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0540faad55b36d54d1c7704ef243b58339e45acd91b6a2bb2fd3fe19a8161f85","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182065,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":1,\"bound\":44}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, b=1, bound=44. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(0, 1, 44, [0])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 44, [0])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0066","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0be04a4480e6784c8ecb6d275764ca01c5b18077a5d9e8d37157b81d43f5a5b","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182066,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":3,\"bound\":45}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, b=3, bound=45. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(1, 3, 45, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 45, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0067","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f49505b2e700af2518b06c24b1b8c72637737f1ed3d3ede6033558d0140ce001","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182067,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":5,\"bound\":46}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, b=5, bound=46. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(2, 5, 46, [0, 10, 20, 30, 40])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 46, [0, 10, 20, 30, 40])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0068","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f2a2f1c85b8ff23c20b21d5249897e24e5d8a50da69910e3cc7fad5212b2690","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182068,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":7,\"bound\":47}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, b=7, bound=47. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(3, 7, 47, [0, 21, 42])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 7, 47, [0, 21, 42])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0069","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce488f69fa34201fa4a4d2adb9f2101c897754ff4fcb6ce88b5a4fbe3157358","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182069,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":4,\"b\":0,\"bound\":48}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, b=0, bound=48. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(4, 0, 48, [0])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, 48, [0])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0070","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51bd520ef0651d5c402b8d6abd178616b51973d1855216d440753374903a4d1f","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182070,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":5,\"b\":2,\"bound\":49}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, b=2, bound=49. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(5, 2, 49, [0, 10, 20, 30, 40])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 2, 49, [0, 10, 20, 30, 40])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0071","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d7f06a016245e07167922f0b686d8bdbc67e31964e21bb8566c8c15e7025a6b","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182071,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":4,\"bound\":50}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=6, b=4, bound=50. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(6, 4, 50, [0, 12, 24, 36, 48])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 4, 50, [0, 12, 24, 36, 48])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0072","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cb1d1f5d6cd0186a268150a580bca486a30490768842a37b4b76a59489c3008","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182072,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":6,\"bound\":51}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, b=6, bound=51. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(0, 6, 51, [0])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 6, 51, [0])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0073","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53ae828f998741e2cd2c0cae626cad39cebeb0f42bc8152ba2fc04cca39ef82c","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182073,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":8,\"bound\":52}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, b=8, bound=52. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(1, 8, 52, [0, 8, 16, 24, 32, 40, 48])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 8, 52, [0, 8, 16, 24, 32, 40, 48])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0074","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfaf557a2c3fe319eda9ce83d5f693cf04462a021b9dd5f0460acc3b1ff7289f","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182074,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":1,\"bound\":53}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, b=1, bound=53. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(2, 1, 53, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 1, 53, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0075","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f63968b687b96df3e3723c3dd4bbe7bda1150c8c00d5359c78da25d00f32aa2","family_id":"qf-ext18-divisibility-common-multiples","seed":202610182075,"semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":3,\"bound\":54}"},"fields":{"title":"All Common Multiples in an Inclusive Natural Window","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, b=3, bound=54. Give (a, b, bound, all common multiples in [0,bound] in increasing order).","answer":{"kind":"expression","value":"(3, 3, 54, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54])","explanation":"Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 3, 54, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54])."},"skills":["All Common Multiples in an Inclusive Natural 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-r02-natural-divisibility-0076","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b5db2de0b76485c80f54ca8f23b270357e3eef1e3bae24212f09908d5916bf6","family_id":"qf-ext18-divisibility-transitivity","seed":202610182076,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":0,\"l\":0}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, k=0, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 0, 0, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0077","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34929bbc5ba621af8e6869deaa76d12888b96543294cf2e7b5ed6c72f5223805","family_id":"qf-ext18-divisibility-transitivity","seed":202610182077,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":1,\"l\":3}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, k=1, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(1, 1, 3, 1, 3, 3, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 1, 3, 1, 3, 3, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0078","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01bef0fd2f77a72c7e0af6072f5a98d70ea848e7f5bb207e2ff192b4a98757d8","family_id":"qf-ext18-divisibility-transitivity","seed":202610182078,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":2,\"l\":6}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, k=2, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(2, 4, 24, 2, 6, 12, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 4, 24, 2, 6, 12, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0079","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f051f3945b662e04698db5efea2b116f0c4c3e40d4eefd47466b017d94c113","family_id":"qf-ext18-divisibility-transitivity","seed":202610182079,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":3,\"l\":2}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, k=3, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(3, 9, 18, 3, 2, 6, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 9, 18, 3, 2, 6, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0080","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dd2f46c1ee901766e51b0b2f4d3ca49d8fffc0b65eabf7e6dbf10f5349acbc8","family_id":"qf-ext18-divisibility-transitivity","seed":202610182080,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":4,\"l\":5}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, k=4, l=5; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(4, 16, 80, 4, 5, 20, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 16, 80, 4, 5, 20, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0081","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1136183f6ae57705a626fc506a60cf2ffed4cede6f1a1a251235688672fdbfd8","family_id":"qf-ext18-divisibility-transitivity","seed":202610182081,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":0,\"l\":1}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, k=0, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(5, 0, 0, 0, 1, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 0, 0, 0, 1, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0082","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70de6593218e31ebaa101064c7b2afbc47883fd1619cb63fa4b8059696a04c55","family_id":"qf-ext18-divisibility-transitivity","seed":202610182082,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":1,\"l\":4}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, k=1, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(0, 0, 0, 1, 4, 4, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 1, 4, 4, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0083","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7537836f3b2d000f86aa671dff847efed1f0968c095351bf2e296b1b27ea34c","family_id":"qf-ext18-divisibility-transitivity","seed":202610182083,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":2,\"l\":0}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, k=2, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(1, 2, 0, 2, 0, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 0, 2, 0, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0084","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1d5eef4871970714710ab0497d7a4b4fd4b2a40d61811634ff55bb75994d026","family_id":"qf-ext18-divisibility-transitivity","seed":202610182084,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":3,\"l\":3}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, k=3, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(2, 6, 18, 3, 3, 9, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 6, 18, 3, 3, 9, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0085","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4e011687a0680aa9a0a291d948ef6161fb032807736aaf84b96cc43b8d94d9f","family_id":"qf-ext18-divisibility-transitivity","seed":202610182085,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":4,\"l\":6}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, k=4, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(3, 12, 72, 4, 6, 24, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 12, 72, 4, 6, 24, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0086","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db7f0fd3c801fc7294437c82f8e241e3ae23eb76682ddf43b21fbcec00bc14ee","family_id":"qf-ext18-divisibility-transitivity","seed":202610182086,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":0,\"l\":2}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, k=0, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(4, 0, 0, 0, 2, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, 0, 0, 2, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0087","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb93deed0fb6d6009fcaf92de053801ed2c4f31dd641e56882080413d1dbdbd1","family_id":"qf-ext18-divisibility-transitivity","seed":202610182087,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":1,\"l\":5}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, k=1, l=5; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(5, 5, 25, 1, 5, 5, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 25, 1, 5, 5, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0088","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7880c0347904ba882dc0e83b848f64a34298ccc23dbf5dbfc560c66e62892750","family_id":"qf-ext18-divisibility-transitivity","seed":202610182088,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":2,\"l\":1}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, k=2, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(0, 0, 0, 2, 1, 2, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 2, 1, 2, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0089","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb1671b46a9c2065d21bc1a8ff1cf62fef0a1402fbaceb9ca7105b83bcf5469c","family_id":"qf-ext18-divisibility-transitivity","seed":202610182089,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":3,\"l\":4}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, k=3, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(1, 3, 12, 3, 4, 12, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 12, 3, 4, 12, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0090","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2843fc3052653ae7c20919172de41b9ceaa776c621df96d0e5abc2d6277a4788","family_id":"qf-ext18-divisibility-transitivity","seed":202610182090,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":4,\"l\":0}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, k=4, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(2, 8, 0, 4, 0, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 8, 0, 4, 0, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0091","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7390d071385ed80c63c54817e1e7eae35ada6498c07159d6aeaf504ee6012646","family_id":"qf-ext18-divisibility-transitivity","seed":202610182091,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":0,\"l\":3}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, k=0, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(3, 0, 0, 0, 3, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 0, 0, 0, 3, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0092","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e38cb6570af4ed6d24c5da86ca9680f767af463127032ce068e52ab4ec272219","family_id":"qf-ext18-divisibility-transitivity","seed":202610182092,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":1,\"l\":6}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, k=1, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(4, 4, 24, 1, 6, 6, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, 24, 1, 6, 6, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0093","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2cd2196cd4fd47894334e151d0e2571d205633e778257f2054ec47920f909e2","family_id":"qf-ext18-divisibility-transitivity","seed":202610182093,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":2,\"l\":2}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, k=2, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(5, 10, 20, 2, 2, 4, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 10, 20, 2, 2, 4, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0094","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"859a201841445a315400a69417e35fd25109f64960aa7b5577cbb64a98763e69","family_id":"qf-ext18-divisibility-transitivity","seed":202610182094,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":3,\"l\":5}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, k=3, l=5; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(0, 0, 0, 3, 5, 15, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 3, 5, 15, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0095","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06f26879beec71d13da3a920b6c4962204319be7f56d53aec42513f6de95a1c2","family_id":"qf-ext18-divisibility-transitivity","seed":202610182095,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":4,\"l\":1}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=1, k=4, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(1, 4, 4, 4, 1, 4, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 4, 4, 4, 1, 4, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0096","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1da0c82ada31da5a9cd8ff0f78f02ac552fd689489bf726238fc5d04f6c9bfa5","family_id":"qf-ext18-divisibility-transitivity","seed":202610182096,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":0,\"l\":4}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=2, k=0, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(2, 0, 0, 0, 4, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 0, 0, 0, 4, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0097","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b52152fe4a69343481f5ca02dfa63cfb89be3be2b0f07bf7a68cd9a2001bae84","family_id":"qf-ext18-divisibility-transitivity","seed":202610182097,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":1,\"l\":0}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=3, k=1, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(3, 3, 0, 1, 0, 0, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 3, 0, 1, 0, 0, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0098","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1595ffdd86df3de904b64200313d954f0ec92153cdadd7e6f3049c0c0cbc883a","family_id":"qf-ext18-divisibility-transitivity","seed":202610182098,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":2,\"l\":3}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=4, k=2, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(4, 8, 24, 2, 3, 6, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 8, 24, 2, 3, 6, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0099","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2edd8ba9286d34dff9968d93f0eb334efe24317efdf2a73af33f3b8fe182ac21","family_id":"qf-ext18-divisibility-transitivity","seed":202610182099,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":3,\"l\":6}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=5, k=3, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(5, 15, 90, 3, 6, 18, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 15, 90, 3, 6, 18, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r02-natural-divisibility-0100","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a968be83d3722dd06363a306d0ae5745113f2741ba46d5d12e73b84f39ed9c9f","family_id":"qf-ext18-divisibility-transitivity","seed":202610182100,"semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":4,\"l\":2}"},"fields":{"title":"Compose Two Given Natural Divisibility Witnesses","stem":"Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=0, k=4, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).","answer":{"kind":"expression","value":"(0, 0, 0, 4, 2, 8, Yes)","explanation":"Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 4, 2, 8, Yes)."},"skills":["Compose Two Given Natural Divisibility Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}