{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0001","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":12,\"b\":18,\"c\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 18, 12, 6, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(12, 18, 12, 6, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066001,"parameters":{"a":12,"b":18,"c":12,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":12,\"b\":18,\"c\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0002","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-18,\"b\":27,\"c\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 27, 19, 9, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-18, 27, 19, 9, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066002,"parameters":{"a":-18,"b":27,"c":19,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-18,\"b\":27,\"c\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0003","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":24,\"b\":36,\"c\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 36, 24, 12, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(24, 36, 24, 12, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066003,"parameters":{"a":24,"b":36,"c":24,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":24,\"b\":36,\"c\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0004","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-30,\"b\":45,\"c\":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-30, 45, 31, 15, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-30, 45, 31, 15, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066004,"parameters":{"a":-30,"b":45,"c":31,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-30,\"b\":45,\"c\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0005","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":36,\"b\":54,\"c\":36}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(36, 54, 36, 18, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(36, 54, 36, 18, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066005,"parameters":{"a":36,"b":54,"c":36,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":36,\"b\":54,\"c\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0006","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-42,\"b\":63,\"c\":43}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-42, 63, 43, 21, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-42, 63, 43, 21, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066006,"parameters":{"a":-42,"b":63,"c":43,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-42,\"b\":63,\"c\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0007","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":48,\"b\":72,\"c\":48}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(48, 72, 48, 24, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(48, 72, 48, 24, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066007,"parameters":{"a":48,"b":72,"c":48,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":48,\"b\":72,\"c\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0008","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-54,\"b\":81,\"c\":55}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-54, 81, 55, 27, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-54, 81, 55, 27, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066008,"parameters":{"a":-54,"b":81,"c":55,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-54,\"b\":81,\"c\":55}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0009","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":60,\"b\":90,\"c\":60}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(60, 90, 60, 30, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(60, 90, 60, 30, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066009,"parameters":{"a":60,"b":90,"c":60,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":60,\"b\":90,\"c\":60}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0010","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-66,\"b\":99,\"c\":67}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-66, 99, 67, 33, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-66, 99, 67, 33, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066010,"parameters":{"a":-66,"b":99,"c":67,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-66,\"b\":99,\"c\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0011","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":72,\"b\":108,\"c\":72}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(72, 108, 72, 36, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(72, 108, 72, 36, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066011,"parameters":{"a":72,"b":108,"c":72,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":72,\"b\":108,\"c\":72}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0012","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-78,\"b\":117,\"c\":79}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-78, 117, 79, 39, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-78, 117, 79, 39, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066012,"parameters":{"a":-78,"b":117,"c":79,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-78,\"b\":117,\"c\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0013","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":84,\"b\":126,\"c\":84}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(84, 126, 84, 42, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(84, 126, 84, 42, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066013,"parameters":{"a":84,"b":126,"c":84,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":84,\"b\":126,\"c\":84}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0014","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-90,\"b\":135,\"c\":91}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-90, 135, 91, 45, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-90, 135, 91, 45, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066014,"parameters":{"a":-90,"b":135,"c":91,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-90,\"b\":135,\"c\":91}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0015","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":96,\"b\":144,\"c\":96}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(96, 144, 96, 48, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(96, 144, 96, 48, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066015,"parameters":{"a":96,"b":144,"c":96,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":96,\"b\":144,\"c\":96}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0016","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-102,\"b\":153,\"c\":103}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-102, 153, 103, 51, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-102, 153, 103, 51, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066016,"parameters":{"a":-102,"b":153,"c":103,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-102,\"b\":153,\"c\":103}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0017","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":108,\"b\":162,\"c\":108}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(108, 162, 108, 54, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(108, 162, 108, 54, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066017,"parameters":{"a":108,"b":162,"c":108,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":108,\"b\":162,\"c\":108}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0018","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-114,\"b\":171,\"c\":115}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-114, 171, 115, 57, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-114, 171, 115, 57, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066018,"parameters":{"a":-114,"b":171,"c":115,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-114,\"b\":171,\"c\":115}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0019","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":120,\"b\":180,\"c\":120}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(120, 180, 120, 60, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(120, 180, 120, 60, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066019,"parameters":{"a":120,"b":180,"c":120,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":120,\"b\":180,\"c\":120}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0020","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-126,\"b\":189,\"c\":127}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-126, 189, 127, 63, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-126, 189, 127, 63, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066020,"parameters":{"a":-126,"b":189,"c":127,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-126,\"b\":189,\"c\":127}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0021","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":132,\"b\":198,\"c\":132}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(132, 198, 132, 66, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(132, 198, 132, 66, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066021,"parameters":{"a":132,"b":198,"c":132,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":132,\"b\":198,\"c\":132}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0022","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-138,\"b\":207,\"c\":139}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-138, 207, 139, 69, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-138, 207, 139, 69, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066022,"parameters":{"a":-138,"b":207,"c":139,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-138,\"b\":207,\"c\":139}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0023","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":144,\"b\":216,\"c\":144}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(144, 216, 144, 72, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(144, 216, 144, 72, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066023,"parameters":{"a":144,"b":216,"c":144,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":144,\"b\":216,\"c\":144}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0024","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":-150,\"b\":225,\"c\":151}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-150, 225, 151, 75, 否, 否, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(-150, 225, 151, 75, 否, 否, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066024,"parameters":{"a":-150,"b":225,"c":151,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":-150,\"b\":225,\"c\":151}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0025","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-solvability；資料={\"a\":156,\"b\":234,\"c\":156}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(156, 234, 156, 78, 否, 是, gcd_divides_c_iff_integer_solution)","explanation_zh":"Wave5封閉層核對=(156, 234, 156, 78, 否, 是, gcd_divides_c_iff_integer_solution)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-solvability","seed":202611066025,"parameters":{"a":156,"b":234,"c":156,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-solvability","semantic_template":"qf-mod37-dioph-solvability/{\"a\":156,\"b\":234,\"c\":156}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0026","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":12,\"b\":18,\"c\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, -1, 1, -1, 1, 6, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(6, -1, 1, -1, 1, 6, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066026,"parameters":{"a":12,"b":18,"c":6,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":12,\"b\":18,\"c\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0027","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-18,\"b\":27,\"c\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 1, 1, 2, 2, 18, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(9, 1, 1, 2, 2, 18, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066027,"parameters":{"a":-18,"b":27,"c":18,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-18,\"b\":27,\"c\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0028","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":24,\"b\":36,\"c\":36}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, -1, 1, -3, 3, 36, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(12, -1, 1, -3, 3, 36, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066028,"parameters":{"a":24,"b":36,"c":36,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":24,\"b\":36,\"c\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0029","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-30,\"b\":45,\"c\":60}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 1, 1, 4, 4, 60, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(15, 1, 1, 4, 4, 60, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066029,"parameters":{"a":-30,"b":45,"c":60,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-30,\"b\":45,\"c\":60}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0030","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":36,\"b\":54,\"c\":90}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, -1, 1, -5, 5, 90, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(18, -1, 1, -5, 5, 90, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066030,"parameters":{"a":36,"b":54,"c":90,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":36,\"b\":54,\"c\":90}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0031","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-42,\"b\":63,\"c\":126}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 1, 1, 6, 6, 126, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(21, 1, 1, 6, 6, 126, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066031,"parameters":{"a":-42,"b":63,"c":126,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-42,\"b\":63,\"c\":126}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0032","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":48,\"b\":72,\"c\":168}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, -1, 1, -7, 7, 168, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(24, -1, 1, -7, 7, 168, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066032,"parameters":{"a":48,"b":72,"c":168,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":48,\"b\":72,\"c\":168}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0033","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-54,\"b\":81,\"c\":216}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 1, 1, 8, 8, 216, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(27, 1, 1, 8, 8, 216, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066033,"parameters":{"a":-54,"b":81,"c":216,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-54,\"b\":81,\"c\":216}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0034","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":60,\"b\":90,\"c\":270}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, -1, 1, -9, 9, 270, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(30, -1, 1, -9, 9, 270, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066034,"parameters":{"a":60,"b":90,"c":270,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":60,\"b\":90,\"c\":270}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0035","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-66,\"b\":99,\"c\":330}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(33, 1, 1, 10, 10, 330, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(33, 1, 1, 10, 10, 330, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066035,"parameters":{"a":-66,"b":99,"c":330,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-66,\"b\":99,\"c\":330}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0036","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":72,\"b\":108,\"c\":396}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(36, -1, 1, -11, 11, 396, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(36, -1, 1, -11, 11, 396, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066036,"parameters":{"a":72,"b":108,"c":396,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":72,\"b\":108,\"c\":396}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0037","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-78,\"b\":117,\"c\":468}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(39, 1, 1, 12, 12, 468, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(39, 1, 1, 12, 12, 468, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066037,"parameters":{"a":-78,"b":117,"c":468,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-78,\"b\":117,\"c\":468}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0038","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":84,\"b\":126,\"c\":546}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(42, -1, 1, -13, 13, 546, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(42, -1, 1, -13, 13, 546, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066038,"parameters":{"a":84,"b":126,"c":546,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":84,\"b\":126,\"c\":546}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0039","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-90,\"b\":135,\"c\":630}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(45, 1, 1, 14, 14, 630, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(45, 1, 1, 14, 14, 630, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066039,"parameters":{"a":-90,"b":135,"c":630,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-90,\"b\":135,\"c\":630}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0040","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":96,\"b\":144,\"c\":720}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(48, -1, 1, -15, 15, 720, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(48, -1, 1, -15, 15, 720, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066040,"parameters":{"a":96,"b":144,"c":720,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":96,\"b\":144,\"c\":720}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0041","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-102,\"b\":153,\"c\":816}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(51, 1, 1, 16, 16, 816, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(51, 1, 1, 16, 16, 816, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066041,"parameters":{"a":-102,"b":153,"c":816,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-102,\"b\":153,\"c\":816}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0042","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":108,\"b\":162,\"c\":918}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(54, -1, 1, -17, 17, 918, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(54, -1, 1, -17, 17, 918, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066042,"parameters":{"a":108,"b":162,"c":918,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":108,\"b\":162,\"c\":918}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0043","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-114,\"b\":171,\"c\":1026}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(57, 1, 1, 18, 18, 1026, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(57, 1, 1, 18, 18, 1026, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066043,"parameters":{"a":-114,"b":171,"c":1026,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-114,\"b\":171,\"c\":1026}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0044","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":120,\"b\":180,\"c\":1140}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(60, -1, 1, -19, 19, 1140, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(60, -1, 1, -19, 19, 1140, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066044,"parameters":{"a":120,"b":180,"c":1140,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":120,\"b\":180,\"c\":1140}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0045","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-126,\"b\":189,\"c\":1260}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(63, 1, 1, 20, 20, 1260, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(63, 1, 1, 20, 20, 1260, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066045,"parameters":{"a":-126,"b":189,"c":1260,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-126,\"b\":189,\"c\":1260}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0046","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":132,\"b\":198,\"c\":1386}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(66, -1, 1, -21, 21, 1386, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(66, -1, 1, -21, 21, 1386, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066046,"parameters":{"a":132,"b":198,"c":1386,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":132,\"b\":198,\"c\":1386}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0047","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-138,\"b\":207,\"c\":1518}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(69, 1, 1, 22, 22, 1518, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(69, 1, 1, 22, 22, 1518, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066047,"parameters":{"a":-138,"b":207,"c":1518,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-138,\"b\":207,\"c\":1518}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0048","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":144,\"b\":216,\"c\":1656}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(72, -1, 1, -23, 23, 1656, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(72, -1, 1, -23, 23, 1656, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066048,"parameters":{"a":144,"b":216,"c":1656,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":144,\"b\":216,\"c\":1656}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0049","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":-150,\"b\":225,\"c\":1800}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(75, 1, 1, 24, 24, 1800, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(75, 1, 1, 24, 24, 1800, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066049,"parameters":{"a":-150,"b":225,"c":1800,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-150,\"b\":225,\"c\":1800}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0050","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-particular-solution；資料={\"a\":156,\"b\":234,\"c\":1950}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(78, -1, 1, -25, 25, 1950, 是, extended_Euclidean_Bezout_certificate)","explanation_zh":"Wave5封閉層核對=(78, -1, 1, -25, 25, 1950, 是, extended_Euclidean_Bezout_certificate)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-particular-solution","seed":202611066050,"parameters":{"a":156,"b":234,"c":1950,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-particular-solution","semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":156,\"b\":234,\"c\":1950}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0051","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":12,\"b\":18,\"c\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 1, 3, -2, x=-1+(3)t, y=1+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-1, 1, 3, -2, x=-1+(3)t, y=1+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066051,"parameters":{"a":12,"b":18,"c":6,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":12,\"b\":18,\"c\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0052","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-18,\"b\":27,\"c\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 3, 2, x=2+(3)t, y=2+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(2, 2, 3, 2, x=2+(3)t, y=2+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066052,"parameters":{"a":-18,"b":27,"c":18,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-18,\"b\":27,\"c\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0053","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":24,\"b\":36,\"c\":36}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 3, 3, -2, x=-3+(3)t, y=3+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-3, 3, 3, -2, x=-3+(3)t, y=3+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066053,"parameters":{"a":24,"b":36,"c":36,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":24,\"b\":36,\"c\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0054","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-30,\"b\":45,\"c\":60}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, 2, x=4+(3)t, y=4+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(4, 4, 3, 2, x=4+(3)t, y=4+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066054,"parameters":{"a":-30,"b":45,"c":60,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-30,\"b\":45,\"c\":60}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0055","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":36,\"b\":54,\"c\":90}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 5, 3, -2, x=-5+(3)t, y=5+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-5, 5, 3, -2, x=-5+(3)t, y=5+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066055,"parameters":{"a":36,"b":54,"c":90,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":36,\"b\":54,\"c\":90}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0056","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-42,\"b\":63,\"c\":126}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 3, 2, x=6+(3)t, y=6+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(6, 6, 3, 2, x=6+(3)t, y=6+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066056,"parameters":{"a":-42,"b":63,"c":126,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-42,\"b\":63,\"c\":126}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0057","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":48,\"b\":72,\"c\":168}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 7, 3, -2, x=-7+(3)t, y=7+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-7, 7, 3, -2, x=-7+(3)t, y=7+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066057,"parameters":{"a":48,"b":72,"c":168,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":48,\"b\":72,\"c\":168}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0058","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-54,\"b\":81,\"c\":216}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 3, 2, x=8+(3)t, y=8+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(8, 8, 3, 2, x=8+(3)t, y=8+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066058,"parameters":{"a":-54,"b":81,"c":216,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-54,\"b\":81,\"c\":216}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0059","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":60,\"b\":90,\"c\":270}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 9, 3, -2, x=-9+(3)t, y=9+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-9, 9, 3, -2, x=-9+(3)t, y=9+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066059,"parameters":{"a":60,"b":90,"c":270,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":60,\"b\":90,\"c\":270}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0060","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-66,\"b\":99,\"c\":330}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 3, 2, x=10+(3)t, y=10+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(10, 10, 3, 2, x=10+(3)t, y=10+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066060,"parameters":{"a":-66,"b":99,"c":330,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-66,\"b\":99,\"c\":330}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0061","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":72,\"b\":108,\"c\":396}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 11, 3, -2, x=-11+(3)t, y=11+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-11, 11, 3, -2, x=-11+(3)t, y=11+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066061,"parameters":{"a":72,"b":108,"c":396,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":72,\"b\":108,\"c\":396}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0062","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-78,\"b\":117,\"c\":468}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 12, 3, 2, x=12+(3)t, y=12+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(12, 12, 3, 2, x=12+(3)t, y=12+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066062,"parameters":{"a":-78,"b":117,"c":468,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-78,\"b\":117,\"c\":468}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0063","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":84,\"b\":126,\"c\":546}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 13, 3, -2, x=-13+(3)t, y=13+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-13, 13, 3, -2, x=-13+(3)t, y=13+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066063,"parameters":{"a":84,"b":126,"c":546,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":84,\"b\":126,\"c\":546}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0064","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-90,\"b\":135,\"c\":630}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066064,"parameters":{"a":-90,"b":135,"c":630,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-90,\"b\":135,\"c\":630}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0065","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":96,\"b\":144,\"c\":720}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 15, 3, -2, x=-15+(3)t, y=15+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-15, 15, 3, -2, x=-15+(3)t, y=15+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066065,"parameters":{"a":96,"b":144,"c":720,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":96,\"b\":144,\"c\":720}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0066","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-102,\"b\":153,\"c\":816}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, 3, 2, x=16+(3)t, y=16+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(16, 16, 3, 2, x=16+(3)t, y=16+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066066,"parameters":{"a":-102,"b":153,"c":816,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-102,\"b\":153,\"c\":816}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0067","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":108,\"b\":162,\"c\":918}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 17, 3, -2, x=-17+(3)t, y=17+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-17, 17, 3, -2, x=-17+(3)t, y=17+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066067,"parameters":{"a":108,"b":162,"c":918,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":108,\"b\":162,\"c\":918}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0068","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-114,\"b\":171,\"c\":1026}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, 3, 2, x=18+(3)t, y=18+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(18, 18, 3, 2, x=18+(3)t, y=18+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066068,"parameters":{"a":-114,"b":171,"c":1026,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-114,\"b\":171,\"c\":1026}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0069","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":120,\"b\":180,\"c\":1140}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 19, 3, -2, x=-19+(3)t, y=19+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-19, 19, 3, -2, x=-19+(3)t, y=19+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066069,"parameters":{"a":120,"b":180,"c":1140,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":120,\"b\":180,\"c\":1140}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0070","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-126,\"b\":189,\"c\":1260}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 20, 3, 2, x=20+(3)t, y=20+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(20, 20, 3, 2, x=20+(3)t, y=20+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066070,"parameters":{"a":-126,"b":189,"c":1260,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-126,\"b\":189,\"c\":1260}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0071","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":132,\"b\":198,\"c\":1386}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 21, 3, -2, x=-21+(3)t, y=21+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-21, 21, 3, -2, x=-21+(3)t, y=21+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066071,"parameters":{"a":132,"b":198,"c":1386,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":132,\"b\":198,\"c\":1386}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0072","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-138,\"b\":207,\"c\":1518}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 22, 3, 2, x=22+(3)t, y=22+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(22, 22, 3, 2, x=22+(3)t, y=22+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066072,"parameters":{"a":-138,"b":207,"c":1518,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-138,\"b\":207,\"c\":1518}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0073","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":144,\"b\":216,\"c\":1656}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 23, 3, -2, x=-23+(3)t, y=23+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-23, 23, 3, -2, x=-23+(3)t, y=23+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066073,"parameters":{"a":144,"b":216,"c":1656,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":144,\"b\":216,\"c\":1656}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0074","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":-150,\"b\":225,\"c\":1800}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 24, 3, 2, x=24+(3)t, y=24+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(24, 24, 3, 2, x=24+(3)t, y=24+(2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066074,"parameters":{"a":-150,"b":225,"c":1800,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-150,\"b\":225,\"c\":1800}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0075","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-general-solution；資料={\"a\":156,\"b\":234,\"c\":1950}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 25, 3, -2, x=-25+(3)t, y=25+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation_zh":"Wave5封閉層核對=(-25, 25, 3, -2, x=-25+(3)t, y=25+(-2)t, all_and_only_integer_solutions_for_t_in_Z)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-general-solution","seed":202611066075,"parameters":{"a":156,"b":234,"c":1950,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-general-solution","semantic_template":"qf-mod37-dioph-general-solution/{\"a\":156,\"b\":234,\"c\":1950}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0076","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-12, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066076,"parameters":{"a":0,"b":0,"c":-12,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0077","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-11, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066077,"parameters":{"a":0,"b":0,"c":-11,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0078","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-10, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066078,"parameters":{"a":0,"b":0,"c":-10,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0079","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-9, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066079,"parameters":{"a":0,"b":0,"c":-9,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0080","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-8, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066080,"parameters":{"a":0,"b":0,"c":-8,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0081","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-7, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066081,"parameters":{"a":0,"b":0,"c":-7,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0082","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-6, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066082,"parameters":{"a":0,"b":0,"c":-6,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0083","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-5, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066083,"parameters":{"a":0,"b":0,"c":-5,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0084","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-4, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066084,"parameters":{"a":0,"b":0,"c":-4,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0087","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(-1, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066087,"parameters":{"a":0,"b":0,"c":-1,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0091","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(3, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066091,"parameters":{"a":0,"b":0,"c":3,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0092","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(4, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066092,"parameters":{"a":0,"b":0,"c":4,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0098","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(10, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066098,"parameters":{"a":0,"b":0,"c":10,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0099","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(11, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066099,"parameters":{"a":0,"b":0,"c":11,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0100","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-diophantine-equation"],"title_zh":"線性Diophantine方程","stem_zh":"exact integer coefficients、normalized gcd與extended Euclidean證書。操作=dioph-degenerate-boundary；資料={\"a\":0,\"b\":0,\"c\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 否, no_integer_solution, never_divide_by_gcd_zero)","explanation_zh":"Wave5封閉層核對=(12, 否, no_integer_solution, never_divide_by_gcd_zero)；依完整明示資料獨立重算；非退化ax+by=c有解iff gcd(a,b)整除c，完整解需Z參數化而非有限搜尋；a=b=0須分開處理且不得除以gcd零；保留零係數與負係數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性Diophantine方程"]},"variant":{"family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066100,"parameters":{"a":0,"b":0,"c":12,"domain":"Exact_integer_linear_Diophantine_equation","operation":"dioph-degenerate-boundary","semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}