# Euler定理基礎 · 脈衝 R06

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations · 2026-10-14 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(5, 2, 4, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0001/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 2. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(6, 5, 2, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0002/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 3. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(7, 4, 6, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0003/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 4. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(8, 5, 4, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0004/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 5. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(9, 7, 6, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0005/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 6. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(10, 7, 4, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0006/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 7. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(11, 8, 10, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0007/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 8. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(12, 11, 4, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0008/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 9. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(13, 10, 12, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0009/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 10. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(14, 11, 6, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0010/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 11. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(15, 13, 8, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0011/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 12. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(16, 13, 8, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0012/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 13. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(17, 14, 16, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0013/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 14. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(18, 17, 6, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0014/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 15. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(19, 16, 18, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0015/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 16. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(20, 17, 8, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0016/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 17. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(21, 19, 12, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0017/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 18. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(22, 19, 10, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0018/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 19. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(23, 20, 22, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0019/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 20. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(24, 23, 8, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0020/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 21. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(25, 22, 20, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0021/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 22. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(26, 23, 12, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0022/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 23. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(27, 25, 18, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0023/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 24. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(28, 25, 12, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0024/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 25. Euler定理

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 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。

查看答案與解釋(29, 26, 28, 1, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0025/)

Euler定理 · qf-mod38-eulerthm-congruence · candidate





## 26. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 2, 3, 4], [1, 2, 3, 4], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0026/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 27. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 5], [1, 5], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0027/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 28. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0028/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 29. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 3, 5, 7], [1, 3, 5, 7], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0029/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 30. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0030/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 31. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 3, 7, 9], [1, 3, 7, 9], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0031/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 32. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0032/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 33. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 5, 7, 11], [1, 5, 7, 11], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0033/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 34. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0034/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 35. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0035/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 36. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0036/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 37. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0037/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 38. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0038/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 39. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0039/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 40. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0040/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 41. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0041/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 42. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0042/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 43. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0043/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 44. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0044/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 45. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], 是, multiplication_by_a_permutes_reduced_residues)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0045/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 46. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0046/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 47. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0047/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 48. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0048/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 49. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0049/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 50. Euler定理

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 [units, sorted_mapped_units, permutation, certificate] 欄序回答。

查看答案與解釋([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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0050/)

Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate





## 51. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-2, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0051/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 52. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-5, 1, 2, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0052/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 53. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-4, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0053/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 54. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-5, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0054/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 55. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-7, 2, 6, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0055/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 56. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-7, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0056/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 57. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-8, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0057/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 58. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-11, 1, 4, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0058/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 59. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-10, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0059/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 60. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-11, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0060/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 61. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-13, 2, 8, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0061/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 62. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-13, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0062/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 63. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-14, 3, 16, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0063/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 64. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-17, 1, 6, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0064/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 65. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-16, 3, 18, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0065/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 66. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-17, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0066/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 67. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-19, 2, 12, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0067/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 68. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-19, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0068/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 69. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-20, 3, 22, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0069/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 70. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-23, 1, 8, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0070/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 71. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-22, 3, 20, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0071/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 72. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-23, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0072/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 73. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-25, 2, 18, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0073/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 74. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-25, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0074/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 75. Euler定理

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 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。

查看答案與解釋(-26, 3, 28, 1, 是, negative_base_normalized_before_exact_modular_power)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0075/)

Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate





## 76. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(6, 6, 6, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0076/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 77. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(7, 7, 7, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0077/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 78. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(8, 8, 8, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0078/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 79. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(9, 9, 9, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0079/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 80. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(10, 10, 10, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0080/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 81. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(11, 11, 11, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0081/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 82. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(12, 12, 12, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0082/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 83. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(13, 13, 13, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0083/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 84. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(14, 14, 14, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0084/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 85. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(15, 15, 15, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0085/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 86. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(16, 16, 16, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0086/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 87. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(17, 17, 17, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0087/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 88. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(18, 18, 18, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0088/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 89. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(19, 19, 19, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0089/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 90. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(20, 20, 20, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0090/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 91. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(21, 21, 21, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0091/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 92. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(22, 22, 22, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0092/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 93. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(23, 23, 23, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0093/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 94. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(24, 24, 24, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0094/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 95. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(25, 25, 25, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0095/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 96. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(26, 26, 26, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0096/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 97. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(27, 27, 27, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0097/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 98. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(28, 28, 28, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0098/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 99. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(29, 29, 29, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0099/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate





## 100. Euler定理

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 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。

查看答案與解釋(30, 30, 30, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0100/)

Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations/validation-report.json)