{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261018-r05-modular-congruence-0001","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":7,\"b\":12}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(0, 7, 12, 否, 不求值, 不求值)","explanation_zh":"擴展基礎核對=(0, 7, 12, 否, 不求值, 不求值)；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185001,"parameters":{"n":0,"a":7,"b":12,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":7,\"b\":12,\"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-r05-modular-congruence-0002","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":1,\"a\":10,\"b\":17}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(1, 10, 17, 是, [0, 0], 是)","explanation_zh":"擴展基礎核對=(1, 10, 17, 是, [0, 0], 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185002,"parameters":{"n":1,"a":10,"b":17,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":10,\"b\":17,\"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-r05-modular-congruence-0003","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":2,\"a\":13,\"b\":22}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(2, 13, 22, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(2, 13, 22, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185003,"parameters":{"n":2,"a":13,"b":22,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":13,\"b\":22,\"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-r05-modular-congruence-0004","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":3,\"a\":16,\"b\":27}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(3, 16, 27, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(3, 16, 27, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185004,"parameters":{"n":3,"a":16,"b":27,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":16,\"b\":27,\"n\":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-r05-modular-congruence-0005","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":4,\"a\":19,\"b\":32}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(4, 19, 32, 是, [3, 0], 否)","explanation_zh":"擴展基礎核對=(4, 19, 32, 是, [3, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185005,"parameters":{"n":4,"a":19,"b":32,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":19,\"b\":32,\"n\":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-r05-modular-congruence-0006","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":5,\"a\":22,\"b\":37}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(5, 22, 37, 是, [2, 2], 是)","explanation_zh":"擴展基礎核對=(5, 22, 37, 是, [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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185006,"parameters":{"n":5,"a":22,"b":37,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":22,\"b\":37,\"n\":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-r05-modular-congruence-0007","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":25,\"b\":42}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(0, 25, 42, 否, 不求值, 不求值)","explanation_zh":"擴展基礎核對=(0, 25, 42, 否, 不求值, 不求值)；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185007,"parameters":{"n":0,"a":25,"b":42,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":25,\"b\":42,\"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-r05-modular-congruence-0008","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":1,\"a\":28,\"b\":47}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(1, 28, 47, 是, [0, 0], 是)","explanation_zh":"擴展基礎核對=(1, 28, 47, 是, [0, 0], 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185008,"parameters":{"n":1,"a":28,"b":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":28,\"b\":47,\"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-r05-modular-congruence-0009","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":2,\"a\":31,\"b\":52}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(2, 31, 52, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(2, 31, 52, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185009,"parameters":{"n":2,"a":31,"b":52,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":31,\"b\":52,\"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-r05-modular-congruence-0010","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":3,\"a\":34,\"b\":57}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(3, 34, 57, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(3, 34, 57, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185010,"parameters":{"n":3,"a":34,"b":57,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":34,\"b\":57,\"n\":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-r05-modular-congruence-0011","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":4,\"a\":37,\"b\":62}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(4, 37, 62, 是, [1, 2], 否)","explanation_zh":"擴展基礎核對=(4, 37, 62, 是, [1, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185011,"parameters":{"n":4,"a":37,"b":62,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":37,\"b\":62,\"n\":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-r05-modular-congruence-0012","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":5,\"a\":40,\"b\":67}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(5, 40, 67, 是, [0, 2], 否)","explanation_zh":"擴展基礎核對=(5, 40, 67, 是, [0, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185012,"parameters":{"n":5,"a":40,"b":67,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":40,\"b\":67,\"n\":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-r05-modular-congruence-0013","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":43,\"b\":72}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(0, 43, 72, 否, 不求值, 不求值)","explanation_zh":"擴展基礎核對=(0, 43, 72, 否, 不求值, 不求值)；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185013,"parameters":{"n":0,"a":43,"b":72,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":43,\"b\":72,\"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-r05-modular-congruence-0014","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":1,\"a\":46,\"b\":77}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(1, 46, 77, 是, [0, 0], 是)","explanation_zh":"擴展基礎核對=(1, 46, 77, 是, [0, 0], 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185014,"parameters":{"n":1,"a":46,"b":77,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":46,\"b\":77,\"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-r05-modular-congruence-0015","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":2,\"a\":49,\"b\":82}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(2, 49, 82, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(2, 49, 82, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185015,"parameters":{"n":2,"a":49,"b":82,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":49,\"b\":82,\"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-r05-modular-congruence-0016","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":3,\"a\":52,\"b\":87}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(3, 52, 87, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(3, 52, 87, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185016,"parameters":{"n":3,"a":52,"b":87,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":52,\"b\":87,\"n\":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-r05-modular-congruence-0017","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":4,\"a\":55,\"b\":92}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(4, 55, 92, 是, [3, 0], 否)","explanation_zh":"擴展基礎核對=(4, 55, 92, 是, [3, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185017,"parameters":{"n":4,"a":55,"b":92,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":55,\"b\":92,\"n\":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-r05-modular-congruence-0018","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":5,\"a\":58,\"b\":97}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(5, 58, 97, 是, [3, 2], 否)","explanation_zh":"擴展基礎核對=(5, 58, 97, 是, [3, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185018,"parameters":{"n":5,"a":58,"b":97,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":58,\"b\":97,\"n\":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-r05-modular-congruence-0019","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":61,\"b\":102}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(0, 61, 102, 否, 不求值, 不求值)","explanation_zh":"擴展基礎核對=(0, 61, 102, 否, 不求值, 不求值)；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185019,"parameters":{"n":0,"a":61,"b":102,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":61,\"b\":102,\"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-r05-modular-congruence-0020","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":1,\"a\":64,\"b\":107}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(1, 64, 107, 是, [0, 0], 是)","explanation_zh":"擴展基礎核對=(1, 64, 107, 是, [0, 0], 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185020,"parameters":{"n":1,"a":64,"b":107,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":64,\"b\":107,\"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-r05-modular-congruence-0021","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":2,\"a\":67,\"b\":112}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(2, 67, 112, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(2, 67, 112, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185021,"parameters":{"n":2,"a":67,"b":112,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":67,\"b\":112,\"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-r05-modular-congruence-0022","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":3,\"a\":70,\"b\":117}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(3, 70, 117, 是, [1, 0], 否)","explanation_zh":"擴展基礎核對=(3, 70, 117, 是, [1, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185022,"parameters":{"n":3,"a":70,"b":117,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":70,\"b\":117,\"n\":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-r05-modular-congruence-0023","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":4,\"a\":73,\"b\":122}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(4, 73, 122, 是, [1, 2], 否)","explanation_zh":"擴展基礎核對=(4, 73, 122, 是, [1, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185023,"parameters":{"n":4,"a":73,"b":122,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":73,\"b\":122,\"n\":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-r05-modular-congruence-0024","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":5,\"a\":76,\"b\":127}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(5, 76, 127, 是, [1, 2], 否)","explanation_zh":"擴展基礎核對=(5, 76, 127, 是, [1, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185024,"parameters":{"n":5,"a":76,"b":127,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":76,\"b\":127,\"n\":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-r05-modular-congruence-0025","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-remainder-gate；資料={\"n\":0,\"a\":79,\"b\":132}。請依 [n,a,b,positive_modulus,remainders_or_不求值,congruent_or_不求值] 的欄序回答。","answer":{"kind":"expression","value":"(0, 79, 132, 否, 不求值, 不求值)","explanation_zh":"擴展基礎核對=(0, 79, 132, 否, 不求值, 不求值)；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-remainder-gate","seed":202610185025,"parameters":{"n":0,"a":79,"b":132,"domain":"Nat_positive_mod_congruence","operation":"modeq-remainder-gate","semantic_template":"qf-ext18-modeq-remainder-gate/{\"a\":79,\"b\":132,\"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-r05-modular-congruence-0026","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":2,\"a\":1,\"bound\":25}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])","explanation_zh":"擴展基礎核對=(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185026,"parameters":{"n":2,"a":1,"bound":25,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":1,\"bound\":25,\"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-r05-modular-congruence-0027","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":3,\"a\":4,\"bound\":26}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(3, 4, 26, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25])","explanation_zh":"擴展基礎核對=(3, 4, 26, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185027,"parameters":{"n":3,"a":4,"bound":26,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":4,\"bound\":26,\"n\":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-r05-modular-congruence-0028","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":4,\"a\":7,\"bound\":27}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(4, 7, 27, 3, [3, 7, 11, 15, 19, 23, 27])","explanation_zh":"擴展基礎核對=(4, 7, 27, 3, [3, 7, 11, 15, 19, 23, 27])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185028,"parameters":{"n":4,"a":7,"bound":27,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":7,\"bound\":27,\"n\":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-r05-modular-congruence-0029","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":5,\"a\":10,\"bound\":28}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(5, 10, 28, 0, [0, 5, 10, 15, 20, 25])","explanation_zh":"擴展基礎核對=(5, 10, 28, 0, [0, 5, 10, 15, 20, 25])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185029,"parameters":{"n":5,"a":10,"bound":28,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":10,\"bound\":28,\"n\":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-r05-modular-congruence-0030","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":6,\"a\":13,\"bound\":29}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(6, 13, 29, 1, [1, 7, 13, 19, 25])","explanation_zh":"擴展基礎核對=(6, 13, 29, 1, [1, 7, 13, 19, 25])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185030,"parameters":{"n":6,"a":13,"bound":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":13,\"bound\":29,\"n\":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-r05-modular-congruence-0031","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":7,\"a\":16,\"bound\":30}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(7, 16, 30, 2, [2, 9, 16, 23, 30])","explanation_zh":"擴展基礎核對=(7, 16, 30, 2, [2, 9, 16, 23, 30])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185031,"parameters":{"n":7,"a":16,"bound":30,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":16,\"bound\":30,\"n\":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-r05-modular-congruence-0032","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":8,\"a\":19,\"bound\":31}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(8, 19, 31, 3, [3, 11, 19, 27])","explanation_zh":"擴展基礎核對=(8, 19, 31, 3, [3, 11, 19, 27])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185032,"parameters":{"n":8,"a":19,"bound":31,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":19,\"bound\":31,\"n\":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-r05-modular-congruence-0033","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":9,\"a\":22,\"bound\":32}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(9, 22, 32, 4, [4, 13, 22, 31])","explanation_zh":"擴展基礎核對=(9, 22, 32, 4, [4, 13, 22, 31])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185033,"parameters":{"n":9,"a":22,"bound":32,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":22,\"bound\":32,\"n\":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-r05-modular-congruence-0034","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":10,\"a\":25,\"bound\":33}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(10, 25, 33, 5, [5, 15, 25])","explanation_zh":"擴展基礎核對=(10, 25, 33, 5, [5, 15, 25])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185034,"parameters":{"n":10,"a":25,"bound":33,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":25,\"bound\":33,\"n\":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-r05-modular-congruence-0035","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":2,\"a\":28,\"bound\":34}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(2, 28, 34, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34])","explanation_zh":"擴展基礎核對=(2, 28, 34, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185035,"parameters":{"n":2,"a":28,"bound":34,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":28,\"bound\":34,\"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-r05-modular-congruence-0036","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":3,\"a\":31,\"bound\":35}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34])","explanation_zh":"擴展基礎核對=(3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185036,"parameters":{"n":3,"a":31,"bound":35,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":31,\"bound\":35,\"n\":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-r05-modular-congruence-0037","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":4,\"a\":34,\"bound\":36}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(4, 34, 36, 2, [2, 6, 10, 14, 18, 22, 26, 30, 34])","explanation_zh":"擴展基礎核對=(4, 34, 36, 2, [2, 6, 10, 14, 18, 22, 26, 30, 34])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185037,"parameters":{"n":4,"a":34,"bound":36,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":34,\"bound\":36,\"n\":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-r05-modular-congruence-0038","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":5,\"a\":37,\"bound\":37}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(5, 37, 37, 2, [2, 7, 12, 17, 22, 27, 32, 37])","explanation_zh":"擴展基礎核對=(5, 37, 37, 2, [2, 7, 12, 17, 22, 27, 32, 37])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185038,"parameters":{"n":5,"a":37,"bound":37,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":37,\"bound\":37,\"n\":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-r05-modular-congruence-0039","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":6,\"a\":40,\"bound\":38}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(6, 40, 38, 4, [4, 10, 16, 22, 28, 34])","explanation_zh":"擴展基礎核對=(6, 40, 38, 4, [4, 10, 16, 22, 28, 34])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185039,"parameters":{"n":6,"a":40,"bound":38,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":40,\"bound\":38,\"n\":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-r05-modular-congruence-0040","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":7,\"a\":43,\"bound\":39}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(7, 43, 39, 1, [1, 8, 15, 22, 29, 36])","explanation_zh":"擴展基礎核對=(7, 43, 39, 1, [1, 8, 15, 22, 29, 36])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185040,"parameters":{"n":7,"a":43,"bound":39,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":43,\"bound\":39,\"n\":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-r05-modular-congruence-0041","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":8,\"a\":46,\"bound\":40}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(8, 46, 40, 6, [6, 14, 22, 30, 38])","explanation_zh":"擴展基礎核對=(8, 46, 40, 6, [6, 14, 22, 30, 38])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185041,"parameters":{"n":8,"a":46,"bound":40,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":46,\"bound\":40,\"n\":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-r05-modular-congruence-0042","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":9,\"a\":49,\"bound\":41}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(9, 49, 41, 4, [4, 13, 22, 31, 40])","explanation_zh":"擴展基礎核對=(9, 49, 41, 4, [4, 13, 22, 31, 40])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185042,"parameters":{"n":9,"a":49,"bound":41,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":49,\"bound\":41,\"n\":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-r05-modular-congruence-0043","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":10,\"a\":52,\"bound\":42}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(10, 52, 42, 2, [2, 12, 22, 32, 42])","explanation_zh":"擴展基礎核對=(10, 52, 42, 2, [2, 12, 22, 32, 42])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185043,"parameters":{"n":10,"a":52,"bound":42,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":52,\"bound\":42,\"n\":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-r05-modular-congruence-0044","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":2,\"a\":55,\"bound\":43}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(2, 55, 43, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43])","explanation_zh":"擴展基礎核對=(2, 55, 43, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185044,"parameters":{"n":2,"a":55,"bound":43,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":55,\"bound\":43,\"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-r05-modular-congruence-0045","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":3,\"a\":58,\"bound\":44}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(3, 58, 44, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43])","explanation_zh":"擴展基礎核對=(3, 58, 44, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185045,"parameters":{"n":3,"a":58,"bound":44,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":58,\"bound\":44,\"n\":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-r05-modular-congruence-0046","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":4,\"a\":61,\"bound\":45}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(4, 61, 45, 1, [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45])","explanation_zh":"擴展基礎核對=(4, 61, 45, 1, [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185046,"parameters":{"n":4,"a":61,"bound":45,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":61,\"bound\":45,\"n\":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-r05-modular-congruence-0047","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":5,\"a\":64,\"bound\":46}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44])","explanation_zh":"擴展基礎核對=(5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185047,"parameters":{"n":5,"a":64,"bound":46,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":64,\"bound\":46,\"n\":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-r05-modular-congruence-0048","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":6,\"a\":67,\"bound\":47}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(6, 67, 47, 1, [1, 7, 13, 19, 25, 31, 37, 43])","explanation_zh":"擴展基礎核對=(6, 67, 47, 1, [1, 7, 13, 19, 25, 31, 37, 43])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185048,"parameters":{"n":6,"a":67,"bound":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":67,\"bound\":47,\"n\":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-r05-modular-congruence-0049","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":7,\"a\":70,\"bound\":48}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(7, 70, 48, 0, [0, 7, 14, 21, 28, 35, 42])","explanation_zh":"擴展基礎核對=(7, 70, 48, 0, [0, 7, 14, 21, 28, 35, 42])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185049,"parameters":{"n":7,"a":70,"bound":48,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":70,\"bound\":48,\"n\":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-r05-modular-congruence-0050","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-equivalence-class；資料={\"n\":8,\"a\":73,\"bound\":49}。請依 [n,a,bound,remainder,class_members] 的欄序回答。","answer":{"kind":"expression","value":"(8, 73, 49, 1, [1, 9, 17, 25, 33, 41, 49])","explanation_zh":"擴展基礎核對=(8, 73, 49, 1, [1, 9, 17, 25, 33, 41, 49])；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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-equivalence-class","seed":202610185050,"parameters":{"n":8,"a":73,"bound":49,"domain":"Nat_positive_mod_congruence","operation":"modeq-equivalence-class","semantic_template":"qf-ext18-modeq-equivalence-class/{\"a\":73,\"bound\":49,\"n\":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-r05-modular-congruence-0051","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":2,\"a\":1,\"b\":3,\"c\":3,\"d\":7}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(2, [1, 3], [3, 7], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(2, [1, 3], [3, 7], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185051,"parameters":{"n":2,"a":1,"b":3,"c":3,"d":7,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":1,\"b\":3,\"c\":3,\"d\":7,\"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-r05-modular-congruence-0052","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":3,\"a\":2,\"b\":8,\"c\":5,\"d\":14}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(3, [2, 8], [5, 14], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(3, [2, 8], [5, 14], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185052,"parameters":{"n":3,"a":2,"b":8,"c":5,"d":14,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":2,\"b\":8,\"c\":5,\"d\":14,\"n\":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-r05-modular-congruence-0053","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":4,\"a\":3,\"b\":15,\"c\":7,\"d\":11}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(4, [3, 15], [7, 11], 是, [2, 2], 是, 是)","explanation_zh":"擴展基礎核對=(4, [3, 15], [7, 11], 是, [2, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185053,"parameters":{"n":4,"a":3,"b":15,"c":7,"d":11,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":3,\"b\":15,\"c\":7,\"d\":11,\"n\":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-r05-modular-congruence-0054","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":5,\"a\":4,\"b\":9,\"c\":9,\"d\":19}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(5, [4, 9], [9, 19], 是, [3, 3], 是, 是)","explanation_zh":"擴展基礎核對=(5, [4, 9], [9, 19], 是, [3, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185054,"parameters":{"n":5,"a":4,"b":9,"c":9,"d":19,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":4,\"b\":9,\"c\":9,\"d\":19,\"n\":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-r05-modular-congruence-0055","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":6,\"a\":5,\"b\":17,\"c\":11,\"d\":29}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(6, [5, 17], [11, 29], 是, [4, 4], 是, 是)","explanation_zh":"擴展基礎核對=(6, [5, 17], [11, 29], 是, [4, 4], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185055,"parameters":{"n":6,"a":5,"b":17,"c":11,"d":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":5,\"b\":17,\"c\":11,\"d\":29,\"n\":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-r05-modular-congruence-0056","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":7,\"a\":6,\"b\":27,\"c\":13,\"d\":20}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(7, [6, 27], [13, 20], 是, [5, 5], 是, 是)","explanation_zh":"擴展基礎核對=(7, [6, 27], [13, 20], 是, [5, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185056,"parameters":{"n":7,"a":6,"b":27,"c":13,"d":20,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":6,\"b\":27,\"c\":13,\"d\":20,\"n\":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-r05-modular-congruence-0057","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":8,\"a\":7,\"b\":15,\"c\":15,\"d\":31}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(8, [7, 15], [15, 31], 是, [6, 6], 是, 是)","explanation_zh":"擴展基礎核對=(8, [7, 15], [15, 31], 是, [6, 6], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185057,"parameters":{"n":8,"a":7,"b":15,"c":15,"d":31,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":7,\"b\":15,\"c\":15,\"d\":31,\"n\":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-r05-modular-congruence-0058","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":9,\"a\":8,\"b\":26,\"c\":17,\"d\":44}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(9, [8, 26], [17, 44], 是, [7, 7], 是, 是)","explanation_zh":"擴展基礎核對=(9, [8, 26], [17, 44], 是, [7, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185058,"parameters":{"n":9,"a":8,"b":26,"c":17,"d":44,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":8,\"b\":26,\"c\":17,\"d\":44,\"n\":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-r05-modular-congruence-0059","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":10,\"a\":9,\"b\":39,\"c\":19,\"d\":29}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(10, [9, 39], [19, 29], 是, [8, 8], 是, 是)","explanation_zh":"擴展基礎核對=(10, [9, 39], [19, 29], 是, [8, 8], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185059,"parameters":{"n":10,"a":9,"b":39,"c":19,"d":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":9,\"b\":39,\"c\":19,\"d\":29,\"n\":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-r05-modular-congruence-0060","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":2,\"a\":10,\"b\":12,\"c\":21,\"d\":25}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(2, [10, 12], [21, 25], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(2, [10, 12], [21, 25], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185060,"parameters":{"n":2,"a":10,"b":12,"c":21,"d":25,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":10,\"b\":12,\"c\":21,\"d\":25,\"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-r05-modular-congruence-0061","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":3,\"a\":11,\"b\":17,\"c\":23,\"d\":32}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(3, [11, 17], [23, 32], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(3, [11, 17], [23, 32], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185061,"parameters":{"n":3,"a":11,"b":17,"c":23,"d":32,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":11,\"b\":17,\"c\":23,\"d\":32,\"n\":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-r05-modular-congruence-0062","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":4,\"a\":12,\"b\":24,\"c\":25,\"d\":29}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(4, [12, 24], [25, 29], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(4, [12, 24], [25, 29], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185062,"parameters":{"n":4,"a":12,"b":24,"c":25,"d":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":12,\"b\":24,\"c\":25,\"d\":29,\"n\":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-r05-modular-congruence-0063","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":5,\"a\":13,\"b\":18,\"c\":27,\"d\":37}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(5, [13, 18], [27, 37], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(5, [13, 18], [27, 37], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185063,"parameters":{"n":5,"a":13,"b":18,"c":27,"d":37,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":13,\"b\":18,\"c\":27,\"d\":37,\"n\":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-r05-modular-congruence-0064","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":6,\"a\":14,\"b\":26,\"c\":29,\"d\":47}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(6, [14, 26], [29, 47], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(6, [14, 26], [29, 47], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185064,"parameters":{"n":6,"a":14,"b":26,"c":29,"d":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":14,\"b\":26,\"c\":29,\"d\":47,\"n\":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-r05-modular-congruence-0065","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":7,\"a\":15,\"b\":36,\"c\":31,\"d\":38}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(7, [15, 36], [31, 38], 是, [4, 4], 是, 是)","explanation_zh":"擴展基礎核對=(7, [15, 36], [31, 38], 是, [4, 4], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185065,"parameters":{"n":7,"a":15,"b":36,"c":31,"d":38,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":15,\"b\":36,\"c\":31,\"d\":38,\"n\":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-r05-modular-congruence-0066","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":8,\"a\":16,\"b\":24,\"c\":33,\"d\":49}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(8, [16, 24], [33, 49], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(8, [16, 24], [33, 49], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185066,"parameters":{"n":8,"a":16,"b":24,"c":33,"d":49,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":16,\"b\":24,\"c\":33,\"d\":49,\"n\":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-r05-modular-congruence-0067","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":9,\"a\":17,\"b\":35,\"c\":35,\"d\":62}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(9, [17, 35], [35, 62], 是, [7, 7], 是, 是)","explanation_zh":"擴展基礎核對=(9, [17, 35], [35, 62], 是, [7, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185067,"parameters":{"n":9,"a":17,"b":35,"c":35,"d":62,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":17,\"b\":35,\"c\":35,\"d\":62,\"n\":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-r05-modular-congruence-0068","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":10,\"a\":18,\"b\":48,\"c\":37,\"d\":47}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(10, [18, 48], [37, 47], 是, [5, 5], 是, 是)","explanation_zh":"擴展基礎核對=(10, [18, 48], [37, 47], 是, [5, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185068,"parameters":{"n":10,"a":18,"b":48,"c":37,"d":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":18,\"b\":48,\"c\":37,\"d\":47,\"n\":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-r05-modular-congruence-0069","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":2,\"a\":19,\"b\":21,\"c\":39,\"d\":43}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(2, [19, 21], [39, 43], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(2, [19, 21], [39, 43], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185069,"parameters":{"n":2,"a":19,"b":21,"c":39,"d":43,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":19,\"b\":21,\"c\":39,\"d\":43,\"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-r05-modular-congruence-0070","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":3,\"a\":20,\"b\":26,\"c\":41,\"d\":50}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(3, [20, 26], [41, 50], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(3, [20, 26], [41, 50], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185070,"parameters":{"n":3,"a":20,"b":26,"c":41,"d":50,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":20,\"b\":26,\"c\":41,\"d\":50,\"n\":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-r05-modular-congruence-0071","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":4,\"a\":21,\"b\":33,\"c\":43,\"d\":47}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(4, [21, 33], [43, 47], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(4, [21, 33], [43, 47], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185071,"parameters":{"n":4,"a":21,"b":33,"c":43,"d":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":21,\"b\":33,\"c\":43,\"d\":47,\"n\":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-r05-modular-congruence-0072","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":5,\"a\":22,\"b\":27,\"c\":45,\"d\":55}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(5, [22, 27], [45, 55], 是, [2, 2], 是, 是)","explanation_zh":"擴展基礎核對=(5, [22, 27], [45, 55], 是, [2, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185072,"parameters":{"n":5,"a":22,"b":27,"c":45,"d":55,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":22,\"b\":27,\"c\":45,\"d\":55,\"n\":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-r05-modular-congruence-0073","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":6,\"a\":23,\"b\":35,\"c\":47,\"d\":65}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(6, [23, 35], [47, 65], 是, [4, 4], 是, 是)","explanation_zh":"擴展基礎核對=(6, [23, 35], [47, 65], 是, [4, 4], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185073,"parameters":{"n":6,"a":23,"b":35,"c":47,"d":65,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":23,\"b\":35,\"c\":47,\"d\":65,\"n\":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-r05-modular-congruence-0074","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":7,\"a\":24,\"b\":45,\"c\":49,\"d\":56}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(7, [24, 45], [49, 56], 是, [3, 3], 是, 是)","explanation_zh":"擴展基礎核對=(7, [24, 45], [49, 56], 是, [3, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185074,"parameters":{"n":7,"a":24,"b":45,"c":49,"d":56,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":24,\"b\":45,\"c\":49,\"d\":56,\"n\":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-r05-modular-congruence-0075","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-add-preservation；資料={\"n\":8,\"a\":25,\"b\":33,\"c\":51,\"d\":67}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(8, [25, 33], [51, 67], 是, [4, 4], 是, 是)","explanation_zh":"擴展基礎核對=(8, [25, 33], [51, 67], 是, [4, 4], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-add-preservation","seed":202610185075,"parameters":{"n":8,"a":25,"b":33,"c":51,"d":67,"domain":"Nat_positive_mod_congruence","operation":"modeq-add-preservation","semantic_template":"qf-ext18-modeq-add-preservation/{\"a\":25,\"b\":33,\"c\":51,\"d\":67,\"n\":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-r05-modular-congruence-0076","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":2,\"a\":1,\"b\":3,\"c\":3,\"d\":7}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(2, [1, 3], [3, 7], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(2, [1, 3], [3, 7], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185076,"parameters":{"n":2,"a":1,"b":3,"c":3,"d":7,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":1,\"b\":3,\"c\":3,\"d\":7,\"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-r05-modular-congruence-0077","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":3,\"a\":2,\"b\":8,\"c\":5,\"d\":14}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(3, [2, 8], [5, 14], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(3, [2, 8], [5, 14], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185077,"parameters":{"n":3,"a":2,"b":8,"c":5,"d":14,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":2,\"b\":8,\"c\":5,\"d\":14,\"n\":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-r05-modular-congruence-0078","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":4,\"a\":3,\"b\":15,\"c\":7,\"d\":11}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(4, [3, 15], [7, 11], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(4, [3, 15], [7, 11], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185078,"parameters":{"n":4,"a":3,"b":15,"c":7,"d":11,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":3,\"b\":15,\"c\":7,\"d\":11,\"n\":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-r05-modular-congruence-0079","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":5,\"a\":4,\"b\":9,\"c\":9,\"d\":19}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(5, [4, 9], [9, 19], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(5, [4, 9], [9, 19], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185079,"parameters":{"n":5,"a":4,"b":9,"c":9,"d":19,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":4,\"b\":9,\"c\":9,\"d\":19,\"n\":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-r05-modular-congruence-0080","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":6,\"a\":5,\"b\":17,\"c\":11,\"d\":29}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(6, [5, 17], [11, 29], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(6, [5, 17], [11, 29], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185080,"parameters":{"n":6,"a":5,"b":17,"c":11,"d":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":5,\"b\":17,\"c\":11,\"d\":29,\"n\":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-r05-modular-congruence-0081","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":7,\"a\":6,\"b\":27,\"c\":13,\"d\":20}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(7, [6, 27], [13, 20], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(7, [6, 27], [13, 20], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185081,"parameters":{"n":7,"a":6,"b":27,"c":13,"d":20,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":6,\"b\":27,\"c\":13,\"d\":20,\"n\":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-r05-modular-congruence-0082","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":8,\"a\":7,\"b\":15,\"c\":15,\"d\":31}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(8, [7, 15], [15, 31], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(8, [7, 15], [15, 31], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185082,"parameters":{"n":8,"a":7,"b":15,"c":15,"d":31,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":7,\"b\":15,\"c\":15,\"d\":31,\"n\":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-r05-modular-congruence-0083","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":9,\"a\":8,\"b\":26,\"c\":17,\"d\":44}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(9, [8, 26], [17, 44], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(9, [8, 26], [17, 44], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185083,"parameters":{"n":9,"a":8,"b":26,"c":17,"d":44,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":8,\"b\":26,\"c\":17,\"d\":44,\"n\":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-r05-modular-congruence-0084","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":10,\"a\":9,\"b\":39,\"c\":19,\"d\":29}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(10, [9, 39], [19, 29], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(10, [9, 39], [19, 29], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185084,"parameters":{"n":10,"a":9,"b":39,"c":19,"d":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":9,\"b\":39,\"c\":19,\"d\":29,\"n\":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-r05-modular-congruence-0085","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":2,\"a\":10,\"b\":12,\"c\":21,\"d\":25}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(2, [10, 12], [21, 25], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(2, [10, 12], [21, 25], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185085,"parameters":{"n":2,"a":10,"b":12,"c":21,"d":25,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":10,\"b\":12,\"c\":21,\"d\":25,\"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-r05-modular-congruence-0086","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":3,\"a\":11,\"b\":17,\"c\":23,\"d\":32}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(3, [11, 17], [23, 32], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(3, [11, 17], [23, 32], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185086,"parameters":{"n":3,"a":11,"b":17,"c":23,"d":32,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":11,\"b\":17,\"c\":23,\"d\":32,\"n\":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-r05-modular-congruence-0087","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":4,\"a\":12,\"b\":24,\"c\":25,\"d\":29}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(4, [12, 24], [25, 29], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(4, [12, 24], [25, 29], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185087,"parameters":{"n":4,"a":12,"b":24,"c":25,"d":29,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":12,\"b\":24,\"c\":25,\"d\":29,\"n\":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-r05-modular-congruence-0088","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":5,\"a\":13,\"b\":18,\"c\":27,\"d\":37}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(5, [13, 18], [27, 37], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(5, [13, 18], [27, 37], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185088,"parameters":{"n":5,"a":13,"b":18,"c":27,"d":37,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":13,\"b\":18,\"c\":27,\"d\":37,\"n\":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-r05-modular-congruence-0089","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":6,\"a\":14,\"b\":26,\"c\":29,\"d\":47}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(6, [14, 26], [29, 47], 是, [4, 4], 是, 是)","explanation_zh":"擴展基礎核對=(6, [14, 26], [29, 47], 是, [4, 4], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185089,"parameters":{"n":6,"a":14,"b":26,"c":29,"d":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":14,\"b\":26,\"c\":29,\"d\":47,\"n\":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-r05-modular-congruence-0090","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":7,\"a\":15,\"b\":36,\"c\":31,\"d\":38}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(7, [15, 36], [31, 38], 是, [3, 3], 是, 是)","explanation_zh":"擴展基礎核對=(7, [15, 36], [31, 38], 是, [3, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185090,"parameters":{"n":7,"a":15,"b":36,"c":31,"d":38,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":15,\"b\":36,\"c\":31,\"d\":38,\"n\":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-r05-modular-congruence-0091","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":8,\"a\":16,\"b\":24,\"c\":33,\"d\":49}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(8, [16, 24], [33, 49], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(8, [16, 24], [33, 49], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185091,"parameters":{"n":8,"a":16,"b":24,"c":33,"d":49,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":16,\"b\":24,\"c\":33,\"d\":49,\"n\":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-r05-modular-congruence-0092","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":9,\"a\":17,\"b\":35,\"c\":35,\"d\":62}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(9, [17, 35], [35, 62], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(9, [17, 35], [35, 62], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185092,"parameters":{"n":9,"a":17,"b":35,"c":35,"d":62,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":17,\"b\":35,\"c\":35,\"d\":62,\"n\":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-r05-modular-congruence-0093","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":10,\"a\":18,\"b\":48,\"c\":37,\"d\":47}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(10, [18, 48], [37, 47], 是, [6, 6], 是, 是)","explanation_zh":"擴展基礎核對=(10, [18, 48], [37, 47], 是, [6, 6], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185093,"parameters":{"n":10,"a":18,"b":48,"c":37,"d":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":18,\"b\":48,\"c\":37,\"d\":47,\"n\":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-r05-modular-congruence-0094","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":2,\"a\":19,\"b\":21,\"c\":39,\"d\":43}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(2, [19, 21], [39, 43], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(2, [19, 21], [39, 43], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185094,"parameters":{"n":2,"a":19,"b":21,"c":39,"d":43,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":19,\"b\":21,\"c\":39,\"d\":43,\"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-r05-modular-congruence-0095","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":3,\"a\":20,\"b\":26,\"c\":41,\"d\":50}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(3, [20, 26], [41, 50], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(3, [20, 26], [41, 50], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185095,"parameters":{"n":3,"a":20,"b":26,"c":41,"d":50,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":20,\"b\":26,\"c\":41,\"d\":50,\"n\":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-r05-modular-congruence-0096","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":4,\"a\":21,\"b\":33,\"c\":43,\"d\":47}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(4, [21, 33], [43, 47], 是, [3, 3], 是, 是)","explanation_zh":"擴展基礎核對=(4, [21, 33], [43, 47], 是, [3, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185096,"parameters":{"n":4,"a":21,"b":33,"c":43,"d":47,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":21,\"b\":33,\"c\":43,\"d\":47,\"n\":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-r05-modular-congruence-0097","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":5,\"a\":22,\"b\":27,\"c\":45,\"d\":55}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(5, [22, 27], [45, 55], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(5, [22, 27], [45, 55], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185097,"parameters":{"n":5,"a":22,"b":27,"c":45,"d":55,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":22,\"b\":27,\"c\":45,\"d\":55,\"n\":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-r05-modular-congruence-0098","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":6,\"a\":23,\"b\":35,\"c\":47,\"d\":65}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(6, [23, 35], [47, 65], 是, [1, 1], 是, 是)","explanation_zh":"擴展基礎核對=(6, [23, 35], [47, 65], 是, [1, 1], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185098,"parameters":{"n":6,"a":23,"b":35,"c":47,"d":65,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":23,\"b\":35,\"c\":47,\"d\":65,\"n\":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-r05-modular-congruence-0099","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":7,\"a\":24,\"b\":45,\"c\":49,\"d\":56}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(7, [24, 45], [49, 56], 是, [0, 0], 是, 是)","explanation_zh":"擴展基礎核對=(7, [24, 45], [49, 56], 是, [0, 0], 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185099,"parameters":{"n":7,"a":24,"b":45,"c":49,"d":56,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":24,\"b\":45,\"c\":49,\"d\":56,\"n\":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-r05-modular-congruence-0100","batch_id":"batch-20261013-advance-20261018-r05-modular-congruence-foundations","language":"zh-Hant","source_mko_ids":["mko-modular-congruence"],"title_zh":"正模數同餘","stem_zh":"固定模數。操作=modeq-mul-preservation；資料={\"n\":8,\"a\":25,\"b\":33,\"c\":51,\"d\":67}。請依 [n,[a,b],[c,d],premises,[result_remainders],conclusion,implication] 的欄序回答。","answer":{"kind":"expression","value":"(8, [25, 33], [51, 67], 是, [3, 3], 是, 是)","explanation_zh":"擴展基礎核對=(8, [25, 33], [51, 67], 是, [3, 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；正模數資格先行；不可無條件取消共同因子。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["正模數同餘"]},"variant":{"family_id":"qf-ext18-modeq-mul-preservation","seed":202610185100,"parameters":{"n":8,"a":25,"b":33,"c":51,"d":67,"domain":"Nat_positive_mod_congruence","operation":"modeq-mul-preservation","semantic_template":"qf-ext18-modeq-mul-preservation/{\"a\":25,\"b\":33,\"c\":51,\"d\":67,\"n\":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"}}