{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r04-prime-number-0001","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":0}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(0, 所有正自然數, 否, 否)","explanation_zh":"擴展基礎核對=(0, 所有正自然數, 否, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184001,"parameters":{"p":0,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0002","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":1}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(1, [1], 否, 否)","explanation_zh":"擴展基礎核對=(1, [1], 否, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184002,"parameters":{"p":1,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0003","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":2}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2], 是, 是)","explanation_zh":"擴展基礎核對=(2, [1, 2], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184003,"parameters":{"p":2,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0004","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":3}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(3, [1, 3], 是, 是)","explanation_zh":"擴展基礎核對=(3, [1, 3], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184004,"parameters":{"p":3,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0005","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":4}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 4], 是, 否)","explanation_zh":"擴展基礎核對=(4, [1, 2, 4], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184005,"parameters":{"p":4,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0006","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":5}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(5, [1, 5], 是, 是)","explanation_zh":"擴展基礎核對=(5, [1, 5], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184006,"parameters":{"p":5,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0007","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":6}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2, 3, 6], 是, 否)","explanation_zh":"擴展基礎核對=(6, [1, 2, 3, 6], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184007,"parameters":{"p":6,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0008","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":7}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(7, [1, 7], 是, 是)","explanation_zh":"擴展基礎核對=(7, [1, 7], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184008,"parameters":{"p":7,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0009","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":8}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(8, [1, 2, 4, 8], 是, 否)","explanation_zh":"擴展基礎核對=(8, [1, 2, 4, 8], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184009,"parameters":{"p":8,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":8}"}},"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-r04-prime-number-0010","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":9}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(9, [1, 3, 9], 是, 否)","explanation_zh":"擴展基礎核對=(9, [1, 3, 9], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184010,"parameters":{"p":9,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":9}"}},"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-r04-prime-number-0011","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":10}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(10, [1, 2, 5, 10], 是, 否)","explanation_zh":"擴展基礎核對=(10, [1, 2, 5, 10], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184011,"parameters":{"p":10,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":10}"}},"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-r04-prime-number-0012","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":11}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(11, [1, 11], 是, 是)","explanation_zh":"擴展基礎核對=(11, [1, 11], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184012,"parameters":{"p":11,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0013","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":12}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(12, [1, 2, 3, 4, 6, 12], 是, 否)","explanation_zh":"擴展基礎核對=(12, [1, 2, 3, 4, 6, 12], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184013,"parameters":{"p":12,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0014","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":13}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(13, [1, 13], 是, 是)","explanation_zh":"擴展基礎核對=(13, [1, 13], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184014,"parameters":{"p":13,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0015","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":14}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(14, [1, 2, 7, 14], 是, 否)","explanation_zh":"擴展基礎核對=(14, [1, 2, 7, 14], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184015,"parameters":{"p":14,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0016","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":15}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(15, [1, 3, 5, 15], 是, 否)","explanation_zh":"擴展基礎核對=(15, [1, 3, 5, 15], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184016,"parameters":{"p":15,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0017","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":16}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(16, [1, 2, 4, 8, 16], 是, 否)","explanation_zh":"擴展基礎核對=(16, [1, 2, 4, 8, 16], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184017,"parameters":{"p":16,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0018","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":17}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(17, [1, 17], 是, 是)","explanation_zh":"擴展基礎核對=(17, [1, 17], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184018,"parameters":{"p":17,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0019","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":18}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(18, [1, 2, 3, 6, 9, 18], 是, 否)","explanation_zh":"擴展基礎核對=(18, [1, 2, 3, 6, 9, 18], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184019,"parameters":{"p":18,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0020","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":19}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(19, [1, 19], 是, 是)","explanation_zh":"擴展基礎核對=(19, [1, 19], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184020,"parameters":{"p":19,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0021","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":20}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(20, [1, 2, 4, 5, 10, 20], 是, 否)","explanation_zh":"擴展基礎核對=(20, [1, 2, 4, 5, 10, 20], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184021,"parameters":{"p":20,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0022","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":21}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(21, [1, 3, 7, 21], 是, 否)","explanation_zh":"擴展基礎核對=(21, [1, 3, 7, 21], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184022,"parameters":{"p":21,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0023","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":22}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(22, [1, 2, 11, 22], 是, 否)","explanation_zh":"擴展基礎核對=(22, [1, 2, 11, 22], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184023,"parameters":{"p":22,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0024","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":23}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(23, [1, 23], 是, 是)","explanation_zh":"擴展基礎核對=(23, [1, 23], 是, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184024,"parameters":{"p":23,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0025","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-divisor-classification；資料={\"p\":24}。請依 [p,positive_divisors_or_all,p_at_least_2,prime] 的欄序回答。","answer":{"kind":"expression","value":"(24, [1, 2, 3, 4, 6, 8, 12, 24], 是, 否)","explanation_zh":"擴展基礎核對=(24, [1, 2, 3, 4, 6, 8, 12, 24], 是, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-divisor-classification","seed":202610184025,"parameters":{"p":24,"domain":"Nat_prime","operation":"prime-divisor-classification","semantic_template":"qf-ext18-prime-divisor-classification/{\"p\":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-r04-prime-number-0026","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":0}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(0, 不適用, 否)","explanation_zh":"擴展基礎核對=(0, 不適用, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184026,"parameters":{"n":0,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0027","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":1}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(1, 不適用, 否)","explanation_zh":"擴展基礎核對=(1, 不適用, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184027,"parameters":{"n":1,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0028","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":2}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是)","explanation_zh":"擴展基礎核對=(2, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184028,"parameters":{"n":2,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0029","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":33}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(33, 3, 否)","explanation_zh":"擴展基礎核對=(33, 3, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184029,"parameters":{"n":33,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0030","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":34}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(34, 2, 否)","explanation_zh":"擴展基礎核對=(34, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184030,"parameters":{"n":34,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0031","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":35}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(35, 5, 否)","explanation_zh":"擴展基礎核對=(35, 5, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184031,"parameters":{"n":35,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0032","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":36}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(36, 2, 否)","explanation_zh":"擴展基礎核對=(36, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184032,"parameters":{"n":36,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0033","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":37}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(37, 37, 是)","explanation_zh":"擴展基礎核對=(37, 37, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184033,"parameters":{"n":37,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0034","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":38}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(38, 2, 否)","explanation_zh":"擴展基礎核對=(38, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184034,"parameters":{"n":38,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0035","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":39}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(39, 3, 否)","explanation_zh":"擴展基礎核對=(39, 3, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184035,"parameters":{"n":39,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0036","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":40}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(40, 2, 否)","explanation_zh":"擴展基礎核對=(40, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184036,"parameters":{"n":40,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0037","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":41}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(41, 41, 是)","explanation_zh":"擴展基礎核對=(41, 41, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184037,"parameters":{"n":41,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0038","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":42}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(42, 2, 否)","explanation_zh":"擴展基礎核對=(42, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184038,"parameters":{"n":42,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0039","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":43}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(43, 43, 是)","explanation_zh":"擴展基礎核對=(43, 43, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184039,"parameters":{"n":43,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0040","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":44}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(44, 2, 否)","explanation_zh":"擴展基礎核對=(44, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184040,"parameters":{"n":44,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0041","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":45}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(45, 3, 否)","explanation_zh":"擴展基礎核對=(45, 3, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184041,"parameters":{"n":45,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0042","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":46}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(46, 2, 否)","explanation_zh":"擴展基礎核對=(46, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184042,"parameters":{"n":46,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0043","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":47}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(47, 47, 是)","explanation_zh":"擴展基礎核對=(47, 47, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184043,"parameters":{"n":47,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0044","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":48}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(48, 2, 否)","explanation_zh":"擴展基礎核對=(48, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184044,"parameters":{"n":48,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0045","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":49}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(49, 7, 否)","explanation_zh":"擴展基礎核對=(49, 7, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184045,"parameters":{"n":49,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0046","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":50}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(50, 2, 否)","explanation_zh":"擴展基礎核對=(50, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184046,"parameters":{"n":50,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0047","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":51}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(51, 3, 否)","explanation_zh":"擴展基礎核對=(51, 3, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184047,"parameters":{"n":51,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0048","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":52}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(52, 2, 否)","explanation_zh":"擴展基礎核對=(52, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184048,"parameters":{"n":52,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0049","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":53}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(53, 53, 是)","explanation_zh":"擴展基礎核對=(53, 53, 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184049,"parameters":{"n":53,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0050","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-smallest-factor；資料={\"n\":54}。請依 [n,smallest_factor_or_不適用,prime] 的欄序回答。","answer":{"kind":"expression","value":"(54, 2, 否)","explanation_zh":"擴展基礎核對=(54, 2, 否)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-smallest-factor","seed":202610184050,"parameters":{"n":54,"domain":"Nat_prime","operation":"prime-smallest-factor","semantic_template":"qf-ext18-prime-smallest-factor/{\"n\":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-r04-prime-number-0051","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":20,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(20, 2, 是, 10, 合數見證成立)","explanation_zh":"擴展基礎核對=(20, 2, 是, 10, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184051,"parameters":{"n":20,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0052","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":22,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(22, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(22, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184052,"parameters":{"n":22,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":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-r04-prime-number-0053","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":24,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(24, 5, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(24, 5, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184053,"parameters":{"n":24,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":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-r04-prime-number-0054","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":26,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(26, 2, 是, 13, 合數見證成立)","explanation_zh":"擴展基礎核對=(26, 2, 是, 13, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184054,"parameters":{"n":26,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0055","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":28,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(28, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(28, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184055,"parameters":{"n":28,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":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-r04-prime-number-0056","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":30,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(30, 5, 是, 6, 合數見證成立)","explanation_zh":"擴展基礎核對=(30, 5, 是, 6, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184056,"parameters":{"n":30,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":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-r04-prime-number-0057","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":32,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(32, 2, 是, 16, 合數見證成立)","explanation_zh":"擴展基礎核對=(32, 2, 是, 16, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184057,"parameters":{"n":32,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0058","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":34,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(34, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(34, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184058,"parameters":{"n":34,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":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-r04-prime-number-0059","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":36,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(36, 5, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(36, 5, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184059,"parameters":{"n":36,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":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-r04-prime-number-0060","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":38,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(38, 2, 是, 19, 合數見證成立)","explanation_zh":"擴展基礎核對=(38, 2, 是, 19, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184060,"parameters":{"n":38,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0061","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":40,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(40, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(40, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184061,"parameters":{"n":40,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":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-r04-prime-number-0062","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":42,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(42, 5, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(42, 5, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184062,"parameters":{"n":42,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":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-r04-prime-number-0063","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":44,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(44, 2, 是, 22, 合數見證成立)","explanation_zh":"擴展基礎核對=(44, 2, 是, 22, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184063,"parameters":{"n":44,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0064","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":46,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(46, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(46, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184064,"parameters":{"n":46,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":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-r04-prime-number-0065","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":48,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(48, 5, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(48, 5, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184065,"parameters":{"n":48,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":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-r04-prime-number-0066","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":50,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(50, 2, 是, 25, 合數見證成立)","explanation_zh":"擴展基礎核對=(50, 2, 是, 25, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184066,"parameters":{"n":50,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0067","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":52,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(52, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(52, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184067,"parameters":{"n":52,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":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-r04-prime-number-0068","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":54,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(54, 5, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(54, 5, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184068,"parameters":{"n":54,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":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-r04-prime-number-0069","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":56,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(56, 2, 是, 28, 合數見證成立)","explanation_zh":"擴展基礎核對=(56, 2, 是, 28, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184069,"parameters":{"n":56,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":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-r04-prime-number-0070","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":58,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(58, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(58, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184070,"parameters":{"n":58,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":58}"}},"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-r04-prime-number-0071","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":60,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(60, 5, 是, 12, 合數見證成立)","explanation_zh":"擴展基礎核對=(60, 5, 是, 12, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184071,"parameters":{"n":60,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":60}"}},"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-r04-prime-number-0072","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":62,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(62, 2, 是, 31, 合數見證成立)","explanation_zh":"擴展基礎核對=(62, 2, 是, 31, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184072,"parameters":{"n":62,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":62}"}},"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-r04-prime-number-0073","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":64,\"d\":3}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(64, 3, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(64, 3, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184073,"parameters":{"n":64,"d":3,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":3,\"n\":64}"}},"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-r04-prime-number-0074","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":66,\"d\":5}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(66, 5, 否, 不適用, 不是非平凡因數見證)","explanation_zh":"擴展基礎核對=(66, 5, 否, 不適用, 不是非平凡因數見證)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184074,"parameters":{"n":66,"d":5,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":5,\"n\":66}"}},"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-r04-prime-number-0075","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-factor-witness；資料={\"n\":68,\"d\":2}。請依 [n,d,valid,quotient_or_不適用,classification] 的欄序回答。","answer":{"kind":"expression","value":"(68, 2, 是, 34, 合數見證成立)","explanation_zh":"擴展基礎核對=(68, 2, 是, 34, 合數見證成立)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-factor-witness","seed":202610184075,"parameters":{"n":68,"d":2,"domain":"Nat_prime","operation":"prime-factor-witness","semantic_template":"qf-ext18-prime-factor-witness/{\"d\":2,\"n\":68}"}},"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-r04-prime-number-0076","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":0,\"hi\":20}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(0, 20, [2, 3, 5, 7, 11, 13, 17, 19], 8)","explanation_zh":"擴展基礎核對=(0, 20, [2, 3, 5, 7, 11, 13, 17, 19], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184076,"parameters":{"lo":0,"hi":20,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":20,\"lo\":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-r04-prime-number-0077","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":5,\"hi\":25}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(5, 25, [5, 7, 11, 13, 17, 19, 23], 7)","explanation_zh":"擴展基礎核對=(5, 25, [5, 7, 11, 13, 17, 19, 23], 7)；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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184077,"parameters":{"lo":5,"hi":25,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":25,\"lo\":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-r04-prime-number-0078","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":10,\"hi\":30}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(10, 30, [11, 13, 17, 19, 23, 29], 6)","explanation_zh":"擴展基礎核對=(10, 30, [11, 13, 17, 19, 23, 29], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184078,"parameters":{"lo":10,"hi":30,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":30,\"lo\":10}"}},"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-r04-prime-number-0079","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":15,\"hi\":35}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(15, 35, [17, 19, 23, 29, 31], 5)","explanation_zh":"擴展基礎核對=(15, 35, [17, 19, 23, 29, 31], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184079,"parameters":{"lo":15,"hi":35,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":35,\"lo\":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-r04-prime-number-0080","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":20,\"hi\":40}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(20, 40, [23, 29, 31, 37], 4)","explanation_zh":"擴展基礎核對=(20, 40, [23, 29, 31, 37], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184080,"parameters":{"lo":20,"hi":40,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":40,\"lo\":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-r04-prime-number-0081","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":25,\"hi\":45}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(25, 45, [29, 31, 37, 41, 43], 5)","explanation_zh":"擴展基礎核對=(25, 45, [29, 31, 37, 41, 43], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184081,"parameters":{"lo":25,"hi":45,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":45,\"lo\":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-r04-prime-number-0082","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":30,\"hi\":50}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(30, 50, [31, 37, 41, 43, 47], 5)","explanation_zh":"擴展基礎核對=(30, 50, [31, 37, 41, 43, 47], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184082,"parameters":{"lo":30,"hi":50,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":50,\"lo\":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-r04-prime-number-0083","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":35,\"hi\":55}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(35, 55, [37, 41, 43, 47, 53], 5)","explanation_zh":"擴展基礎核對=(35, 55, [37, 41, 43, 47, 53], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184083,"parameters":{"lo":35,"hi":55,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":55,\"lo\":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-r04-prime-number-0084","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":40,\"hi\":60}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(40, 60, [41, 43, 47, 53, 59], 5)","explanation_zh":"擴展基礎核對=(40, 60, [41, 43, 47, 53, 59], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184084,"parameters":{"lo":40,"hi":60,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":60,\"lo\":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-r04-prime-number-0085","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":45,\"hi\":65}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(45, 65, [47, 53, 59, 61], 4)","explanation_zh":"擴展基礎核對=(45, 65, [47, 53, 59, 61], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184085,"parameters":{"lo":45,"hi":65,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":65,\"lo\":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-r04-prime-number-0086","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":50,\"hi\":70}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(50, 70, [53, 59, 61, 67], 4)","explanation_zh":"擴展基礎核對=(50, 70, [53, 59, 61, 67], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184086,"parameters":{"lo":50,"hi":70,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":70,\"lo\":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-r04-prime-number-0087","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":55,\"hi\":75}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(55, 75, [59, 61, 67, 71, 73], 5)","explanation_zh":"擴展基礎核對=(55, 75, [59, 61, 67, 71, 73], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184087,"parameters":{"lo":55,"hi":75,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":75,\"lo\":55}"}},"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-r04-prime-number-0088","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":60,\"hi\":80}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(60, 80, [61, 67, 71, 73, 79], 5)","explanation_zh":"擴展基礎核對=(60, 80, [61, 67, 71, 73, 79], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184088,"parameters":{"lo":60,"hi":80,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":80,\"lo\":60}"}},"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-r04-prime-number-0089","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":65,\"hi\":85}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(65, 85, [67, 71, 73, 79, 83], 5)","explanation_zh":"擴展基礎核對=(65, 85, [67, 71, 73, 79, 83], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184089,"parameters":{"lo":65,"hi":85,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":85,\"lo\":65}"}},"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-r04-prime-number-0090","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":70,\"hi\":90}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(70, 90, [71, 73, 79, 83, 89], 5)","explanation_zh":"擴展基礎核對=(70, 90, [71, 73, 79, 83, 89], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184090,"parameters":{"lo":70,"hi":90,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":90,\"lo\":70}"}},"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-r04-prime-number-0091","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":75,\"hi\":95}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(75, 95, [79, 83, 89], 3)","explanation_zh":"擴展基礎核對=(75, 95, [79, 83, 89], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184091,"parameters":{"lo":75,"hi":95,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":95,\"lo\":75}"}},"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-r04-prime-number-0092","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":80,\"hi\":100}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(80, 100, [83, 89, 97], 3)","explanation_zh":"擴展基礎核對=(80, 100, [83, 89, 97], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184092,"parameters":{"lo":80,"hi":100,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":100,\"lo\":80}"}},"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-r04-prime-number-0093","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":85,\"hi\":105}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(85, 105, [89, 97, 101, 103], 4)","explanation_zh":"擴展基礎核對=(85, 105, [89, 97, 101, 103], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184093,"parameters":{"lo":85,"hi":105,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":105,\"lo\":85}"}},"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-r04-prime-number-0094","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":90,\"hi\":110}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(90, 110, [97, 101, 103, 107, 109], 5)","explanation_zh":"擴展基礎核對=(90, 110, [97, 101, 103, 107, 109], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184094,"parameters":{"lo":90,"hi":110,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":110,\"lo\":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-r04-prime-number-0095","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":95,\"hi\":115}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(95, 115, [97, 101, 103, 107, 109, 113], 6)","explanation_zh":"擴展基礎核對=(95, 115, [97, 101, 103, 107, 109, 113], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184095,"parameters":{"lo":95,"hi":115,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":115,\"lo\":95}"}},"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-r04-prime-number-0096","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":100,\"hi\":120}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(100, 120, [101, 103, 107, 109, 113], 5)","explanation_zh":"擴展基礎核對=(100, 120, [101, 103, 107, 109, 113], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184096,"parameters":{"lo":100,"hi":120,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":120,\"lo\":100}"}},"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-r04-prime-number-0097","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":105,\"hi\":125}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(105, 125, [107, 109, 113], 3)","explanation_zh":"擴展基礎核對=(105, 125, [107, 109, 113], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184097,"parameters":{"lo":105,"hi":125,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":125,\"lo\":105}"}},"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-r04-prime-number-0098","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":110,\"hi\":130}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(110, 130, [113, 127], 2)","explanation_zh":"擴展基礎核對=(110, 130, [113, 127], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184098,"parameters":{"lo":110,"hi":130,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":130,\"lo\":110}"}},"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-r04-prime-number-0099","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":115,\"hi\":135}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(115, 135, [127, 131], 2)","explanation_zh":"擴展基礎核對=(115, 135, [127, 131], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184099,"parameters":{"lo":115,"hi":135,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":135,\"lo\":115}"}},"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-r04-prime-number-0100","batch_id":"batch-20261013-advance-20261018-r04-prime-number-foundations","language":"zh-Hant","source_mko_ids":["mko-prime-number"],"title_zh":"自然數質數與因數","stem_zh":"自然數質數。操作=prime-interval-inventory；資料={\"lo\":120,\"hi\":140}。請依 [lo,hi,primes,count] 的欄序回答。","answer":{"kind":"expression","value":"(120, 140, [127, 131, 137, 139], 4)","explanation_zh":"擴展基礎核對=(120, 140, [127, 131, 137, 139], 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與1不是質數；因數檢查必須完整到平方根界。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["自然數質數與因數"]},"variant":{"family_id":"qf-ext18-prime-interval-inventory","seed":202610184100,"parameters":{"lo":120,"hi":140,"domain":"Nat_prime","operation":"prime-interval-inventory","semantic_template":"qf-ext18-prime-interval-inventory/{\"hi\":140,\"lo\":120}"}},"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"}}