{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r06-euler-theorem-number-theory-0001","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 4, 1, 是)","explanation_zh":"Wave6根層核對=(5, 2, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076001,"parameters":{"n":5,"a":2,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":2,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0002","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":6,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 2, 1, 是)","explanation_zh":"Wave6根層核對=(6, 5, 2, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076002,"parameters":{"n":6,"a":5,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":5,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0003","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":7,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 4, 6, 1, 是)","explanation_zh":"Wave6根層核對=(7, 4, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076003,"parameters":{"n":7,"a":4,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":4,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0004","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":8,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 5, 4, 1, 是)","explanation_zh":"Wave6根層核對=(8, 5, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076004,"parameters":{"n":8,"a":5,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":5,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0005","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":9,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 7, 6, 1, 是)","explanation_zh":"Wave6根層核對=(9, 7, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076005,"parameters":{"n":9,"a":7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0006","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":10,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 7, 4, 1, 是)","explanation_zh":"Wave6根層核對=(10, 7, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076006,"parameters":{"n":10,"a":7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0007","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":11,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 8, 10, 1, 是)","explanation_zh":"Wave6根層核對=(11, 8, 10, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076007,"parameters":{"n":11,"a":8,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":8,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0008","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":12,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 11, 4, 1, 是)","explanation_zh":"Wave6根層核對=(12, 11, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076008,"parameters":{"n":12,"a":11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":11,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0009","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":13,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 10, 12, 1, 是)","explanation_zh":"Wave6根層核對=(13, 10, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076009,"parameters":{"n":13,"a":10,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":10,\"n\":13}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0010","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":14,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 11, 6, 1, 是)","explanation_zh":"Wave6根層核對=(14, 11, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076010,"parameters":{"n":14,"a":11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":11,\"n\":14}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0011","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":15,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 13, 8, 1, 是)","explanation_zh":"Wave6根層核對=(15, 13, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076011,"parameters":{"n":15,"a":13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":13,\"n\":15}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0012","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":16,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 13, 8, 1, 是)","explanation_zh":"Wave6根層核對=(16, 13, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076012,"parameters":{"n":16,"a":13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":13,\"n\":16}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0013","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":17,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 14, 16, 1, 是)","explanation_zh":"Wave6根層核對=(17, 14, 16, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076013,"parameters":{"n":17,"a":14,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":14,\"n\":17}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0014","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":18,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 17, 6, 1, 是)","explanation_zh":"Wave6根層核對=(18, 17, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076014,"parameters":{"n":18,"a":17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":17,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0015","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":19,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 16, 18, 1, 是)","explanation_zh":"Wave6根層核對=(19, 16, 18, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076015,"parameters":{"n":19,"a":16,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":16,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0016","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":20,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 17, 8, 1, 是)","explanation_zh":"Wave6根層核對=(20, 17, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076016,"parameters":{"n":20,"a":17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":17,\"n\":20}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0017","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":21,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 19, 12, 1, 是)","explanation_zh":"Wave6根層核對=(21, 19, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076017,"parameters":{"n":21,"a":19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":19,\"n\":21}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0018","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":22,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 19, 10, 1, 是)","explanation_zh":"Wave6根層核對=(22, 19, 10, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076018,"parameters":{"n":22,"a":19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":19,\"n\":22}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0019","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":23,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 20, 22, 1, 是)","explanation_zh":"Wave6根層核對=(23, 20, 22, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076019,"parameters":{"n":23,"a":20,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":20,\"n\":23}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0020","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":24,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 23, 8, 1, 是)","explanation_zh":"Wave6根層核對=(24, 23, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076020,"parameters":{"n":24,"a":23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":23,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0021","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":25,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 22, 20, 1, 是)","explanation_zh":"Wave6根層核對=(25, 22, 20, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076021,"parameters":{"n":25,"a":22,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":22,\"n\":25}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0022","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":26,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 23, 12, 1, 是)","explanation_zh":"Wave6根層核對=(26, 23, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076022,"parameters":{"n":26,"a":23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":23,\"n\":26}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0023","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":27,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 25, 18, 1, 是)","explanation_zh":"Wave6根層核對=(27, 25, 18, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076023,"parameters":{"n":27,"a":25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":25,\"n\":27}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0024","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":28,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(28, 25, 12, 1, 是)","explanation_zh":"Wave6根層核對=(28, 25, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076024,"parameters":{"n":28,"a":25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":25,\"n\":28}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0025","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":29,\"a\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 26, 28, 1, 是)","explanation_zh":"Wave6根層核對=(29, 26, 28, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-congruence","seed":202611076025,"parameters":{"n":29,"a":26,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-congruence","semantic_template":"qf-mod38-eulerthm-congruence/{\"a\":26,\"n\":29}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0026","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4], [1, 2, 3, 4], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4], [1, 2, 3, 4], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076026,"parameters":{"n":5,"a":2,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":2,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0027","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":6,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 5], [1, 5], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 5], [1, 5], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076027,"parameters":{"n":6,"a":5,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":5,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0028","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":7,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076028,"parameters":{"n":7,"a":4,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":4,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0029","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":8,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7], [1, 3, 5, 7], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 5, 7], [1, 3, 5, 7], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076029,"parameters":{"n":8,"a":5,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":5,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0030","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":9,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076030,"parameters":{"n":9,"a":7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0031","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":10,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 7, 9], [1, 3, 7, 9], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 7, 9], [1, 3, 7, 9], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076031,"parameters":{"n":10,"a":7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0032","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":11,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076032,"parameters":{"n":11,"a":8,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":8,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0033","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":12,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 5, 7, 11], [1, 5, 7, 11], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 5, 7, 11], [1, 5, 7, 11], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076033,"parameters":{"n":12,"a":11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":11,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0034","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":13,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076034,"parameters":{"n":13,"a":10,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":10,\"n\":13}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0035","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":14,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076035,"parameters":{"n":14,"a":11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":11,\"n\":14}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0036","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":15,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076036,"parameters":{"n":15,"a":13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":13,\"n\":15}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0037","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":16,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076037,"parameters":{"n":16,"a":13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":13,\"n\":16}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0038","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":17,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076038,"parameters":{"n":17,"a":14,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":14,\"n\":17}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0039","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":18,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076039,"parameters":{"n":18,"a":17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":17,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0040","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":19,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076040,"parameters":{"n":19,"a":16,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":16,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0041","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":20,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076041,"parameters":{"n":20,"a":17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":17,\"n\":20}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0042","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":21,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076042,"parameters":{"n":21,"a":19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":19,\"n\":21}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0043","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":22,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076043,"parameters":{"n":22,"a":19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":19,\"n\":22}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0044","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":23,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076044,"parameters":{"n":23,"a":20,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":20,\"n\":23}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0045","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":24,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076045,"parameters":{"n":24,"a":23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":23,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0046","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":25,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076046,"parameters":{"n":25,"a":22,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":22,\"n\":25}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0047","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":26,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076047,"parameters":{"n":26,"a":23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":23,\"n\":26}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0048","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":27,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076048,"parameters":{"n":27,"a":25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":25,\"n\":27}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0049","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":28,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076049,"parameters":{"n":28,"a":25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":25,\"n\":28}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0050","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={\"n\":29,\"a\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], 是, multiplication_by_a_permutes_reduced_residues)","explanation_zh":"Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-unit-permutation","seed":202611076050,"parameters":{"n":29,"a":26,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-unit-permutation","semantic_template":"qf-mod38-eulerthm-unit-permutation/{\"a\":26,\"n\":29}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0051","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":5,\"a\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-2, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076051,"parameters":{"n":5,"a":-2,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-2,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0052","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":6,\"a\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 1, 2, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-5, 1, 2, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076052,"parameters":{"n":6,"a":-5,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-5,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0053","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":7,\"a\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-4, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076053,"parameters":{"n":7,"a":-4,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-4,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0054","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":8,\"a\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-5, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076054,"parameters":{"n":8,"a":-5,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-5,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0055","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":9,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 2, 6, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-7, 2, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076055,"parameters":{"n":9,"a":-7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0056","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":10,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-7, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076056,"parameters":{"n":10,"a":-7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0057","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":11,\"a\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-8, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076057,"parameters":{"n":11,"a":-8,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-8,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0058","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":12,\"a\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 1, 4, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-11, 1, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076058,"parameters":{"n":12,"a":-11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-11,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0059","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":13,\"a\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-10, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076059,"parameters":{"n":13,"a":-10,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-10,\"n\":13}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0060","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":14,\"a\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-11, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076060,"parameters":{"n":14,"a":-11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-11,\"n\":14}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0061","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":15,\"a\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 2, 8, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-13, 2, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076061,"parameters":{"n":15,"a":-13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-13,\"n\":15}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0062","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":16,\"a\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-13, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076062,"parameters":{"n":16,"a":-13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-13,\"n\":16}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0063","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":17,\"a\":-14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 3, 16, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-14, 3, 16, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076063,"parameters":{"n":17,"a":-14,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-14,\"n\":17}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0064","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":18,\"a\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 1, 6, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-17, 1, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076064,"parameters":{"n":18,"a":-17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-17,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0065","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":19,\"a\":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 3, 18, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-16, 3, 18, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076065,"parameters":{"n":19,"a":-16,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-16,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0066","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":20,\"a\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-17, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076066,"parameters":{"n":20,"a":-17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-17,\"n\":20}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0067","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":21,\"a\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 2, 12, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-19, 2, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076067,"parameters":{"n":21,"a":-19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-19,\"n\":21}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0068","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":22,\"a\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-19, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076068,"parameters":{"n":22,"a":-19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-19,\"n\":22}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0069","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":23,\"a\":-20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 3, 22, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-20, 3, 22, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076069,"parameters":{"n":23,"a":-20,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-20,\"n\":23}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0070","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":24,\"a\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 1, 8, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-23, 1, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076070,"parameters":{"n":24,"a":-23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-23,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0071","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":25,\"a\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 3, 20, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-22, 3, 20, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076071,"parameters":{"n":25,"a":-22,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-22,\"n\":25}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0072","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":26,\"a\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-23, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076072,"parameters":{"n":26,"a":-23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-23,\"n\":26}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0073","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":27,\"a\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 2, 18, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-25, 2, 18, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076073,"parameters":{"n":27,"a":-25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-25,\"n\":27}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0074","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":28,\"a\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-25, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076074,"parameters":{"n":28,"a":-25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-25,\"n\":28}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0075","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={\"n\":29,\"a\":-26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-26, 3, 28, 1, 是, negative_base_normalized_before_exact_modular_power)","explanation_zh":"Wave6根層核對=(-26, 3, 28, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-negative-base-normalization","seed":202611076075,"parameters":{"n":29,"a":-26,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-negative-base-normalization","semantic_template":"qf-mod38-eulerthm-negative-base-normalization/{\"a\":-26,\"n\":29}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0076","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":6,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 6, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(6, 6, 6, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076076,"parameters":{"n":6,"a":6,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":6,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0077","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":7,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 7, 7, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(7, 7, 7, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076077,"parameters":{"n":7,"a":7,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":7,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0078","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":8,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 8, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(8, 8, 8, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076078,"parameters":{"n":8,"a":8,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":8,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0079","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":9,\"a\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 9, 9, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(9, 9, 9, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076079,"parameters":{"n":9,"a":9,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":9,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0080","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":10,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 10, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(10, 10, 10, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076080,"parameters":{"n":10,"a":10,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":10,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0081","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":11,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 11, 11, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(11, 11, 11, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076081,"parameters":{"n":11,"a":11,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":11,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0082","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":12,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 12, 12, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(12, 12, 12, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076082,"parameters":{"n":12,"a":12,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":12,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0083","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":13,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 13, 13, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(13, 13, 13, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076083,"parameters":{"n":13,"a":13,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":13,\"n\":13}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0084","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":14,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 14, 14, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(14, 14, 14, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076084,"parameters":{"n":14,"a":14,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":14,\"n\":14}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0085","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":15,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 15, 15, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(15, 15, 15, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076085,"parameters":{"n":15,"a":15,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":15,\"n\":15}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0086","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":16,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, 16, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(16, 16, 16, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076086,"parameters":{"n":16,"a":16,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":16,\"n\":16}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0087","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":17,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 17, 17, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(17, 17, 17, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076087,"parameters":{"n":17,"a":17,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":17,\"n\":17}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0088","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":18,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, 18, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(18, 18, 18, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076088,"parameters":{"n":18,"a":18,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":18,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0089","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":19,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 19, 19, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(19, 19, 19, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076089,"parameters":{"n":19,"a":19,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":19,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0090","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":20,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 20, 20, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(20, 20, 20, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076090,"parameters":{"n":20,"a":20,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":20,\"n\":20}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0091","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":21,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 21, 21, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(21, 21, 21, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076091,"parameters":{"n":21,"a":21,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":21,\"n\":21}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0092","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":22,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 22, 22, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(22, 22, 22, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076092,"parameters":{"n":22,"a":22,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":22,\"n\":22}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0093","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":23,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 23, 23, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(23, 23, 23, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076093,"parameters":{"n":23,"a":23,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":23,\"n\":23}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0094","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":24,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 24, 24, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(24, 24, 24, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076094,"parameters":{"n":24,"a":24,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":24,\"n\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0095","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":25,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 25, 25, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(25, 25, 25, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076095,"parameters":{"n":25,"a":25,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":25,\"n\":25}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0096","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":26,\"a\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 26, 26, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(26, 26, 26, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076096,"parameters":{"n":26,"a":26,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":26,\"n\":26}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0097","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":27,\"a\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 27, 27, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(27, 27, 27, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076097,"parameters":{"n":27,"a":27,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":27,\"n\":27}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0098","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":28,\"a\":28}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(28, 28, 28, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(28, 28, 28, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076098,"parameters":{"n":28,"a":28,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":28,\"n\":28}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0099","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":29,\"a\":29}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 29, 29, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(29, 29, 29, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076099,"parameters":{"n":29,"a":29,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":29,\"n\":29}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r06-euler-theorem-number-theory-0100","batch_id":"batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-theorem-number-theory"],"title_zh":"Euler定理","stem_zh":"n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={\"n\":30,\"a\":30}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, 30, 30, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)","explanation_zh":"Wave6根層核對=(30, 30, 30, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler定理"]},"variant":{"family_id":"qf-mod38-eulerthm-nonunit-boundary","seed":202611076100,"parameters":{"n":30,"a":30,"domain":"Exact_integer_Euler_theorem","operation":"eulerthm-nonunit-boundary","semantic_template":"qf-mod38-eulerthm-nonunit-boundary/{\"a\":30,\"n\":30}"}},"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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}