第 16644 題：Euler φ函數

原題與已知條件

正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={"n":19}。請依 [n,coprime_residues,representative_convention] 欄序回答。

• G001 — 正整數與完整1到n代表約定。操作=totient-coprime-residues；資料={"n":19}。請依 [n,coprime_residues,representative_convention] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-euler-totient-function

逐步推導

selected modulus representative1:original nonnegative Euclidean inputs=[1, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 19, 0, 1], [19, 1, 19, 0]];terminal positive gcd=1

selected modulus:original representative k=1 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative2:original nonnegative Euclidean inputs=[2, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 19, 0, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal positive gcd=1

selected modulus:original representative k=2 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative3:original nonnegative Euclidean inputs=[3, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 19, 0, 3], [19, 3, 6, 1], [3, 1, 3, 0]];terminal positive gcd=1

selected modulus:original representative k=3 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative4:original nonnegative Euclidean inputs=[4, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 19, 0, 4], [19, 4, 4, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

selected modulus:original representative k=4 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative5:original nonnegative Euclidean inputs=[5, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 19, 0, 5], [19, 5, 3, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

selected modulus:original representative k=5 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative6:original nonnegative Euclidean inputs=[6, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 19, 0, 6], [19, 6, 3, 1], [6, 1, 6, 0]];terminal positive gcd=1

selected modulus:original representative k=6 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative7:original nonnegative Euclidean inputs=[7, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 19, 0, 7], [19, 7, 2, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

selected modulus:original representative k=7 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative8:original nonnegative Euclidean inputs=[8, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 19, 0, 8], [19, 8, 2, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

selected modulus:original representative k=8 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative9:original nonnegative Euclidean inputs=[9, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 19, 0, 9], [19, 9, 2, 1], [9, 1, 9, 0]];terminal positive gcd=1

selected modulus:original representative k=9 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative10:original nonnegative Euclidean inputs=[10, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 19, 0, 10], [19, 10, 1, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal positive gcd=1

selected modulus:original representative k=10 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative11:original nonnegative Euclidean inputs=[11, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 19, 0, 11], [19, 11, 1, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

selected modulus:original representative k=11 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative12:original nonnegative Euclidean inputs=[12, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 19, 0, 12], [19, 12, 1, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

selected modulus:original representative k=12 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative13:original nonnegative Euclidean inputs=[13, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 19, 0, 13], [19, 13, 1, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal positive gcd=1

selected modulus:original representative k=13 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative14:original nonnegative Euclidean inputs=[14, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 19, 0, 14], [19, 14, 1, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

selected modulus:original representative k=14 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative15:original nonnegative Euclidean inputs=[15, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 19, 0, 15], [19, 15, 1, 4], [15, 4, 3, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

selected modulus:original representative k=15 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative16:original nonnegative Euclidean inputs=[16, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 19, 0, 16], [19, 16, 1, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal positive gcd=1

selected modulus:original representative k=16 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative17:original nonnegative Euclidean inputs=[17, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 19, 0, 17], [19, 17, 1, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal positive gcd=1

selected modulus:original representative k=17 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative18:original nonnegative Euclidean inputs=[18, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 19, 0, 18], [19, 18, 1, 1], [18, 1, 18, 0]];terminal positive gcd=1

selected modulus:original representative k=18 in1..19;verified gcd(k,n)=1

coprime indicator=1;include representative=是

selected modulus representative19:original nonnegative Euclidean inputs=[19, 19]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 19, 1, 0]];terminal positive gcd=19

selected modulus:original representative k=19 in1..19;verified gcd(k,n)=19

coprime indicator=0;include representative=否

selected modulus exact coprime indicator count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0)

selected modulus exact coprime indicator count = 18

selected modulus:complete representative range1..19;all selected coprime representatives=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

phi(19)=18;representatives_1_through_n

逐步反向核對

• S040 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S039 · premise_and_result_audit — 以精確有理數重算 selected modulus exact coprime indicator count，不以小數近似替代。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。