{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0001","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":0,\"b\":0}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 是, 0)","explanation_zh":"擴展基礎核對=(0, 0, 是, 0)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182001,"parameters":{"a":0,"b":0,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":0,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0002","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":0,\"b\":7}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(0, 7, 否, 不存在)","explanation_zh":"擴展基礎核對=(0, 7, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182002,"parameters":{"a":0,"b":7,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":0,\"b\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0003","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":2,\"b\":7}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(2, 7, 否, 不存在)","explanation_zh":"擴展基礎核對=(2, 7, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182003,"parameters":{"a":2,"b":7,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":2,\"b\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0004","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":3,\"b\":12}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(3, 12, 是, 4)","explanation_zh":"擴展基礎核對=(3, 12, 是, 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182004,"parameters":{"a":3,"b":12,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":3,\"b\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0005","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":4,\"b\":21}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(4, 21, 否, 不存在)","explanation_zh":"擴展基礎核對=(4, 21, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182005,"parameters":{"a":4,"b":21,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":4,\"b\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0006","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":5,\"b\":30}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(5, 30, 是, 6)","explanation_zh":"擴展基礎核對=(5, 30, 是, 6)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182006,"parameters":{"a":5,"b":30,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":5,\"b\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0007","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":6,\"b\":43}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(6, 43, 否, 不存在)","explanation_zh":"擴展基礎核對=(6, 43, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182007,"parameters":{"a":6,"b":43,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":6,\"b\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0008","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":7,\"b\":56}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(7, 56, 是, 8)","explanation_zh":"擴展基礎核對=(7, 56, 是, 8)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182008,"parameters":{"a":7,"b":56,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":7,\"b\":56}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0009","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":8,\"b\":73}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(8, 73, 否, 不存在)","explanation_zh":"擴展基礎核對=(8, 73, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182009,"parameters":{"a":8,"b":73,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":8,\"b\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0010","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":9,\"b\":90}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(9, 90, 是, 10)","explanation_zh":"擴展基礎核對=(9, 90, 是, 10)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182010,"parameters":{"a":9,"b":90,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":9,\"b\":90}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0011","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":10,\"b\":111}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(10, 111, 否, 不存在)","explanation_zh":"擴展基礎核對=(10, 111, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182011,"parameters":{"a":10,"b":111,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":10,\"b\":111}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0012","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":11,\"b\":132}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(11, 132, 是, 12)","explanation_zh":"擴展基礎核對=(11, 132, 是, 12)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182012,"parameters":{"a":11,"b":132,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":11,\"b\":132}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0013","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":12,\"b\":157}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(12, 157, 否, 不存在)","explanation_zh":"擴展基礎核對=(12, 157, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182013,"parameters":{"a":12,"b":157,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":12,\"b\":157}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0014","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":13,\"b\":182}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(13, 182, 是, 14)","explanation_zh":"擴展基礎核對=(13, 182, 是, 14)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182014,"parameters":{"a":13,"b":182,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":13,\"b\":182}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0015","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":14,\"b\":211}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(14, 211, 否, 不存在)","explanation_zh":"擴展基礎核對=(14, 211, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182015,"parameters":{"a":14,"b":211,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":14,\"b\":211}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0016","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":15,\"b\":240}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(15, 240, 是, 16)","explanation_zh":"擴展基礎核對=(15, 240, 是, 16)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182016,"parameters":{"a":15,"b":240,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":15,\"b\":240}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0017","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":16,\"b\":273}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(16, 273, 否, 不存在)","explanation_zh":"擴展基礎核對=(16, 273, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182017,"parameters":{"a":16,"b":273,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":16,\"b\":273}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0018","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":17,\"b\":306}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(17, 306, 是, 18)","explanation_zh":"擴展基礎核對=(17, 306, 是, 18)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182018,"parameters":{"a":17,"b":306,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":17,\"b\":306}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0019","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":18,\"b\":343}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(18, 343, 否, 不存在)","explanation_zh":"擴展基礎核對=(18, 343, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182019,"parameters":{"a":18,"b":343,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":18,\"b\":343}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0020","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":19,\"b\":380}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(19, 380, 是, 20)","explanation_zh":"擴展基礎核對=(19, 380, 是, 20)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182020,"parameters":{"a":19,"b":380,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":19,\"b\":380}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0021","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":20,\"b\":421}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(20, 421, 否, 不存在)","explanation_zh":"擴展基礎核對=(20, 421, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182021,"parameters":{"a":20,"b":421,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":20,\"b\":421}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0022","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":21,\"b\":462}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(21, 462, 是, 22)","explanation_zh":"擴展基礎核對=(21, 462, 是, 22)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182022,"parameters":{"a":21,"b":462,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":21,\"b\":462}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0023","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":22,\"b\":507}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(22, 507, 否, 不存在)","explanation_zh":"擴展基礎核對=(22, 507, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182023,"parameters":{"a":22,"b":507,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":22,\"b\":507}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0024","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":23,\"b\":552}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(23, 552, 是, 24)","explanation_zh":"擴展基礎核對=(23, 552, 是, 24)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182024,"parameters":{"a":23,"b":552,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":23,\"b\":552}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0025","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-witness；資料={\"a\":24,\"b\":601}。請依 [a,b,divides,witness_or_不存在] 的欄序回答。","answer":{"kind":"expression","value":"(24, 601, 否, 不存在)","explanation_zh":"擴展基礎核對=(24, 601, 否, 不存在)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-witness","seed":202610182025,"parameters":{"a":24,"b":601,"domain":"Nat_divisibility","operation":"divisibility-witness","semantic_template":"qf-ext18-divisibility-witness/{\"a\":24,\"b\":601}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0026","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":0}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(0, 所有正自然數, 無限)","explanation_zh":"擴展基礎核對=(0, 所有正自然數, 無限)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182026,"parameters":{"b":0,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0027","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":11}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(11, [1, 11], 2)","explanation_zh":"擴展基礎核對=(11, [1, 11], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182027,"parameters":{"b":11,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0028","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":12}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(12, [1, 2, 3, 4, 6, 12], 6)","explanation_zh":"擴展基礎核對=(12, [1, 2, 3, 4, 6, 12], 6)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182028,"parameters":{"b":12,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0029","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":13}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(13, [1, 13], 2)","explanation_zh":"擴展基礎核對=(13, [1, 13], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182029,"parameters":{"b":13,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0030","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":14}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(14, [1, 2, 7, 14], 4)","explanation_zh":"擴展基礎核對=(14, [1, 2, 7, 14], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182030,"parameters":{"b":14,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0031","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":15}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(15, [1, 3, 5, 15], 4)","explanation_zh":"擴展基礎核對=(15, [1, 3, 5, 15], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182031,"parameters":{"b":15,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0032","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":16}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(16, [1, 2, 4, 8, 16], 5)","explanation_zh":"擴展基礎核對=(16, [1, 2, 4, 8, 16], 5)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182032,"parameters":{"b":16,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0033","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":17}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(17, [1, 17], 2)","explanation_zh":"擴展基礎核對=(17, [1, 17], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182033,"parameters":{"b":17,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0034","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":18}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(18, [1, 2, 3, 6, 9, 18], 6)","explanation_zh":"擴展基礎核對=(18, [1, 2, 3, 6, 9, 18], 6)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182034,"parameters":{"b":18,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0035","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":19}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(19, [1, 19], 2)","explanation_zh":"擴展基礎核對=(19, [1, 19], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182035,"parameters":{"b":19,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0036","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":20}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(20, [1, 2, 4, 5, 10, 20], 6)","explanation_zh":"擴展基礎核對=(20, [1, 2, 4, 5, 10, 20], 6)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182036,"parameters":{"b":20,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0037","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":21}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(21, [1, 3, 7, 21], 4)","explanation_zh":"擴展基礎核對=(21, [1, 3, 7, 21], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182037,"parameters":{"b":21,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0038","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":22}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(22, [1, 2, 11, 22], 4)","explanation_zh":"擴展基礎核對=(22, [1, 2, 11, 22], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182038,"parameters":{"b":22,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0039","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":23}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(23, [1, 23], 2)","explanation_zh":"擴展基礎核對=(23, [1, 23], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182039,"parameters":{"b":23,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0040","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":24}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(24, [1, 2, 3, 4, 6, 8, 12, 24], 8)","explanation_zh":"擴展基礎核對=(24, [1, 2, 3, 4, 6, 8, 12, 24], 8)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182040,"parameters":{"b":24,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0041","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":25}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(25, [1, 5, 25], 3)","explanation_zh":"擴展基礎核對=(25, [1, 5, 25], 3)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182041,"parameters":{"b":25,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0042","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":26}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(26, [1, 2, 13, 26], 4)","explanation_zh":"擴展基礎核對=(26, [1, 2, 13, 26], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182042,"parameters":{"b":26,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0043","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":27}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(27, [1, 3, 9, 27], 4)","explanation_zh":"擴展基礎核對=(27, [1, 3, 9, 27], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182043,"parameters":{"b":27,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0044","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":28}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(28, [1, 2, 4, 7, 14, 28], 6)","explanation_zh":"擴展基礎核對=(28, [1, 2, 4, 7, 14, 28], 6)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182044,"parameters":{"b":28,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0045","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":29}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(29, [1, 29], 2)","explanation_zh":"擴展基礎核對=(29, [1, 29], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182045,"parameters":{"b":29,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0046","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":30}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)","explanation_zh":"擴展基礎核對=(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182046,"parameters":{"b":30,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0047","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":31}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(31, [1, 31], 2)","explanation_zh":"擴展基礎核對=(31, [1, 31], 2)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182047,"parameters":{"b":31,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0048","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":32}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(32, [1, 2, 4, 8, 16, 32], 6)","explanation_zh":"擴展基礎核對=(32, [1, 2, 4, 8, 16, 32], 6)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182048,"parameters":{"b":32,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0049","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":33}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(33, [1, 3, 11, 33], 4)","explanation_zh":"擴展基礎核對=(33, [1, 3, 11, 33], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182049,"parameters":{"b":33,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0050","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-divisor-set；資料={\"b\":34}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。","answer":{"kind":"expression","value":"(34, [1, 2, 17, 34], 4)","explanation_zh":"擴展基礎核對=(34, [1, 2, 17, 34], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-divisor-set","seed":202610182050,"parameters":{"b":34,"domain":"Nat_divisibility","operation":"divisibility-divisor-set","semantic_template":"qf-ext18-divisibility-divisor-set/{\"b\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0051","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":0,\"b\":0,\"bound\":30}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 30, [0])","explanation_zh":"擴展基礎核對=(0, 0, 30, [0])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182051,"parameters":{"a":0,"b":0,"bound":30,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":0,\"bound\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0052","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":1,\"b\":2,\"bound\":31}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(1, 2, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30])","explanation_zh":"擴展基礎核對=(1, 2, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182052,"parameters":{"a":1,"b":2,"bound":31,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":2,\"bound\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0053","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":2,\"b\":4,\"bound\":32}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(2, 4, 32, [0, 4, 8, 12, 16, 20, 24, 28, 32])","explanation_zh":"擴展基礎核對=(2, 4, 32, [0, 4, 8, 12, 16, 20, 24, 28, 32])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182053,"parameters":{"a":2,"b":4,"bound":32,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":4,\"bound\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0054","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":3,\"b\":6,\"bound\":33}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(3, 6, 33, [0, 6, 12, 18, 24, 30])","explanation_zh":"擴展基礎核對=(3, 6, 33, [0, 6, 12, 18, 24, 30])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182054,"parameters":{"a":3,"b":6,"bound":33,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":6,\"bound\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0055","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":4,\"b\":8,\"bound\":34}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(4, 8, 34, [0, 8, 16, 24, 32])","explanation_zh":"擴展基礎核對=(4, 8, 34, [0, 8, 16, 24, 32])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182055,"parameters":{"a":4,"b":8,"bound":34,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":4,\"b\":8,\"bound\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0056","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":5,\"b\":1,\"bound\":35}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(5, 1, 35, [0, 5, 10, 15, 20, 25, 30, 35])","explanation_zh":"擴展基礎核對=(5, 1, 35, [0, 5, 10, 15, 20, 25, 30, 35])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182056,"parameters":{"a":5,"b":1,"bound":35,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":5,\"b\":1,\"bound\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0057","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":6,\"b\":3,\"bound\":36}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(6, 3, 36, [0, 6, 12, 18, 24, 30, 36])","explanation_zh":"擴展基礎核對=(6, 3, 36, [0, 6, 12, 18, 24, 30, 36])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182057,"parameters":{"a":6,"b":3,"bound":36,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":3,\"bound\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0058","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":0,\"b\":5,\"bound\":37}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(0, 5, 37, [0])","explanation_zh":"擴展基礎核對=(0, 5, 37, [0])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182058,"parameters":{"a":0,"b":5,"bound":37,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":5,\"bound\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0059","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":1,\"b\":7,\"bound\":38}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(1, 7, 38, [0, 7, 14, 21, 28, 35])","explanation_zh":"擴展基礎核對=(1, 7, 38, [0, 7, 14, 21, 28, 35])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182059,"parameters":{"a":1,"b":7,"bound":38,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":7,\"bound\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0060","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":2,\"b\":0,\"bound\":39}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(2, 0, 39, [0])","explanation_zh":"擴展基礎核對=(2, 0, 39, [0])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182060,"parameters":{"a":2,"b":0,"bound":39,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":0,\"bound\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0061","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":3,\"b\":2,\"bound\":40}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(3, 2, 40, [0, 6, 12, 18, 24, 30, 36])","explanation_zh":"擴展基礎核對=(3, 2, 40, [0, 6, 12, 18, 24, 30, 36])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182061,"parameters":{"a":3,"b":2,"bound":40,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":2,\"bound\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0062","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":4,\"b\":4,\"bound\":41}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(4, 4, 41, [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40])","explanation_zh":"擴展基礎核對=(4, 4, 41, [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182062,"parameters":{"a":4,"b":4,"bound":41,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":4,\"b\":4,\"bound\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0063","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":5,\"b\":6,\"bound\":42}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(5, 6, 42, [0, 30])","explanation_zh":"擴展基礎核對=(5, 6, 42, [0, 30])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182063,"parameters":{"a":5,"b":6,"bound":42,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":5,\"b\":6,\"bound\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0064","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":6,\"b\":8,\"bound\":43}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(6, 8, 43, [0, 24])","explanation_zh":"擴展基礎核對=(6, 8, 43, [0, 24])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182064,"parameters":{"a":6,"b":8,"bound":43,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":8,\"bound\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0065","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":0,\"b\":1,\"bound\":44}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(0, 1, 44, [0])","explanation_zh":"擴展基礎核對=(0, 1, 44, [0])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182065,"parameters":{"a":0,"b":1,"bound":44,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":1,\"bound\":44}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0066","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":1,\"b\":3,\"bound\":45}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(1, 3, 45, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45])","explanation_zh":"擴展基礎核對=(1, 3, 45, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182066,"parameters":{"a":1,"b":3,"bound":45,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":3,\"bound\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0067","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":2,\"b\":5,\"bound\":46}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(2, 5, 46, [0, 10, 20, 30, 40])","explanation_zh":"擴展基礎核對=(2, 5, 46, [0, 10, 20, 30, 40])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182067,"parameters":{"a":2,"b":5,"bound":46,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":5,\"bound\":46}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0068","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":3,\"b\":7,\"bound\":47}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(3, 7, 47, [0, 21, 42])","explanation_zh":"擴展基礎核對=(3, 7, 47, [0, 21, 42])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182068,"parameters":{"a":3,"b":7,"bound":47,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":7,\"bound\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0069","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":4,\"b\":0,\"bound\":48}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(4, 0, 48, [0])","explanation_zh":"擴展基礎核對=(4, 0, 48, [0])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182069,"parameters":{"a":4,"b":0,"bound":48,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":4,\"b\":0,\"bound\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0070","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":5,\"b\":2,\"bound\":49}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(5, 2, 49, [0, 10, 20, 30, 40])","explanation_zh":"擴展基礎核對=(5, 2, 49, [0, 10, 20, 30, 40])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182070,"parameters":{"a":5,"b":2,"bound":49,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":5,\"b\":2,\"bound\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0071","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":6,\"b\":4,\"bound\":50}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(6, 4, 50, [0, 12, 24, 36, 48])","explanation_zh":"擴展基礎核對=(6, 4, 50, [0, 12, 24, 36, 48])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182071,"parameters":{"a":6,"b":4,"bound":50,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":4,\"bound\":50}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0072","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":0,\"b\":6,\"bound\":51}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(0, 6, 51, [0])","explanation_zh":"擴展基礎核對=(0, 6, 51, [0])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182072,"parameters":{"a":0,"b":6,"bound":51,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":0,\"b\":6,\"bound\":51}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0073","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":1,\"b\":8,\"bound\":52}。請依 [a,b,bound,common_multiples] 的欄序回答。","answer":{"kind":"expression","value":"(1, 8, 52, [0, 8, 16, 24, 32, 40, 48])","explanation_zh":"擴展基礎核對=(1, 8, 52, [0, 8, 16, 24, 32, 40, 48])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182073,"parameters":{"a":1,"b":8,"bound":52,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":1,\"b\":8,\"bound\":52}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0074","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":2,\"b\":1,\"bound\":53}。請依 [a,b,bound,common_multiples] 的欄序回答。","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_zh":"擴展基礎核對=(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])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182074,"parameters":{"a":2,"b":1,"bound":53,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":2,\"b\":1,\"bound\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0075","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-common-multiples；資料={\"a\":3,\"b\":3,\"bound\":54}。請依 [a,b,bound,common_multiples] 的欄序回答。","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_zh":"擴展基礎核對=(3, 3, 54, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-common-multiples","seed":202610182075,"parameters":{"a":3,"b":3,"bound":54,"domain":"Nat_divisibility","operation":"divisibility-common-multiples","semantic_template":"qf-ext18-divisibility-common-multiples/{\"a\":3,\"b\":3,\"bound\":54}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0076","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":0,\"k\":0,\"l\":0}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0, 0, 是)","explanation_zh":"擴展基礎核對=(0, 0, 0, 0, 0, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182076,"parameters":{"a":0,"k":0,"l":0,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":0,\"l\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0077","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":1,\"k\":1,\"l\":3}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(1, 1, 3, 1, 3, 3, 是)","explanation_zh":"擴展基礎核對=(1, 1, 3, 1, 3, 3, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182077,"parameters":{"a":1,"k":1,"l":3,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":1,\"l\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0078","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":2,\"k\":2,\"l\":6}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(2, 4, 24, 2, 6, 12, 是)","explanation_zh":"擴展基礎核對=(2, 4, 24, 2, 6, 12, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182078,"parameters":{"a":2,"k":2,"l":6,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":2,\"l\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0079","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":3,\"k\":3,\"l\":2}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(3, 9, 18, 3, 2, 6, 是)","explanation_zh":"擴展基礎核對=(3, 9, 18, 3, 2, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182079,"parameters":{"a":3,"k":3,"l":2,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":3,\"l\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0080","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":4,\"k\":4,\"l\":5}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(4, 16, 80, 4, 5, 20, 是)","explanation_zh":"擴展基礎核對=(4, 16, 80, 4, 5, 20, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182080,"parameters":{"a":4,"k":4,"l":5,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":4,\"l\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0081","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":5,\"k\":0,\"l\":1}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(5, 0, 0, 0, 1, 0, 是)","explanation_zh":"擴展基礎核對=(5, 0, 0, 0, 1, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182081,"parameters":{"a":5,"k":0,"l":1,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":0,\"l\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0082","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":0,\"k\":1,\"l\":4}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 1, 4, 4, 是)","explanation_zh":"擴展基礎核對=(0, 0, 0, 1, 4, 4, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182082,"parameters":{"a":0,"k":1,"l":4,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":1,\"l\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0083","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":1,\"k\":2,\"l\":0}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(1, 2, 0, 2, 0, 0, 是)","explanation_zh":"擴展基礎核對=(1, 2, 0, 2, 0, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182083,"parameters":{"a":1,"k":2,"l":0,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":2,\"l\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0084","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":2,\"k\":3,\"l\":3}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(2, 6, 18, 3, 3, 9, 是)","explanation_zh":"擴展基礎核對=(2, 6, 18, 3, 3, 9, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182084,"parameters":{"a":2,"k":3,"l":3,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":3,\"l\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0085","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":3,\"k\":4,\"l\":6}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(3, 12, 72, 4, 6, 24, 是)","explanation_zh":"擴展基礎核對=(3, 12, 72, 4, 6, 24, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182085,"parameters":{"a":3,"k":4,"l":6,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":4,\"l\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0086","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":4,\"k\":0,\"l\":2}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(4, 0, 0, 0, 2, 0, 是)","explanation_zh":"擴展基礎核對=(4, 0, 0, 0, 2, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182086,"parameters":{"a":4,"k":0,"l":2,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":0,\"l\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0087","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":5,\"k\":1,\"l\":5}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(5, 5, 25, 1, 5, 5, 是)","explanation_zh":"擴展基礎核對=(5, 5, 25, 1, 5, 5, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182087,"parameters":{"a":5,"k":1,"l":5,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":1,\"l\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0088","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":0,\"k\":2,\"l\":1}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 2, 1, 2, 是)","explanation_zh":"擴展基礎核對=(0, 0, 0, 2, 1, 2, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182088,"parameters":{"a":0,"k":2,"l":1,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":2,\"l\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0089","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":1,\"k\":3,\"l\":4}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(1, 3, 12, 3, 4, 12, 是)","explanation_zh":"擴展基礎核對=(1, 3, 12, 3, 4, 12, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182089,"parameters":{"a":1,"k":3,"l":4,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":3,\"l\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0090","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":2,\"k\":4,\"l\":0}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(2, 8, 0, 4, 0, 0, 是)","explanation_zh":"擴展基礎核對=(2, 8, 0, 4, 0, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182090,"parameters":{"a":2,"k":4,"l":0,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":4,\"l\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0091","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":3,\"k\":0,\"l\":3}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(3, 0, 0, 0, 3, 0, 是)","explanation_zh":"擴展基礎核對=(3, 0, 0, 0, 3, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182091,"parameters":{"a":3,"k":0,"l":3,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":0,\"l\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0092","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":4,\"k\":1,\"l\":6}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(4, 4, 24, 1, 6, 6, 是)","explanation_zh":"擴展基礎核對=(4, 4, 24, 1, 6, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182092,"parameters":{"a":4,"k":1,"l":6,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":1,\"l\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0093","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":5,\"k\":2,\"l\":2}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(5, 10, 20, 2, 2, 4, 是)","explanation_zh":"擴展基礎核對=(5, 10, 20, 2, 2, 4, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182093,"parameters":{"a":5,"k":2,"l":2,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":2,\"l\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0094","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":0,\"k\":3,\"l\":5}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 3, 5, 15, 是)","explanation_zh":"擴展基礎核對=(0, 0, 0, 3, 5, 15, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182094,"parameters":{"a":0,"k":3,"l":5,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":3,\"l\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0095","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":1,\"k\":4,\"l\":1}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(1, 4, 4, 4, 1, 4, 是)","explanation_zh":"擴展基礎核對=(1, 4, 4, 4, 1, 4, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182095,"parameters":{"a":1,"k":4,"l":1,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":1,\"k\":4,\"l\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0096","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":2,\"k\":0,\"l\":4}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(2, 0, 0, 0, 4, 0, 是)","explanation_zh":"擴展基礎核對=(2, 0, 0, 0, 4, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182096,"parameters":{"a":2,"k":0,"l":4,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":2,\"k\":0,\"l\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0097","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":3,\"k\":1,\"l\":0}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(3, 3, 0, 1, 0, 0, 是)","explanation_zh":"擴展基礎核對=(3, 3, 0, 1, 0, 0, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182097,"parameters":{"a":3,"k":1,"l":0,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":3,\"k\":1,\"l\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0098","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":4,\"k\":2,\"l\":3}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(4, 8, 24, 2, 3, 6, 是)","explanation_zh":"擴展基礎核對=(4, 8, 24, 2, 3, 6, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182098,"parameters":{"a":4,"k":2,"l":3,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":4,\"k\":2,\"l\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0099","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":5,\"k\":3,\"l\":6}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(5, 15, 90, 3, 6, 18, 是)","explanation_zh":"擴展基礎核對=(5, 15, 90, 3, 6, 18, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182099,"parameters":{"a":5,"k":3,"l":6,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":5,\"k\":3,\"l\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r02-natural-divisibility-0100","batch_id":"batch-20261013-advance-20261018-r02-natural-divisibility-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-divisibility"],"title_zh":"自然數整除與零邊界","stem_zh":"自然數包含0。操作=divisibility-transitivity；資料={\"a\":0,\"k\":4,\"l\":2}。請依 [a,b,c,k,l,k*l,a_divides_c] 的欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 4, 2, 8, 是)","explanation_zh":"擴展基礎核對=(0, 0, 0, 4, 2, 8, 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0∣b當且僅當b=0；a∣0對所有自然數a成立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數整除與零邊界"]},"variant":{"family_id":"qf-ext18-divisibility-transitivity","seed":202610182100,"parameters":{"a":0,"k":4,"l":2,"domain":"Nat_divisibility","operation":"divisibility-transitivity","semantic_template":"qf-ext18-divisibility-transitivity/{\"a\":0,\"k\":4,\"l\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}