第 16653 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-multiplicativity-instance；資料={"m":2,"n":7}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。

• G001 — 正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={"m":2,"n":7}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 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

totient multiplicativity coprimality premise:original nonnegative Euclidean inputs=[2, 7]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 7, 0, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

totient multiplicativity original modulus product := mul(2, 7)

totient multiplicativity original modulus product = 14

left modulus representative1:original nonnegative Euclidean inputs=[1, 2]

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

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

coprime indicator=1;include representative=是

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

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

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

coprime indicator=0;include representative=否

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

left modulus exact coprime indicator count = 1

left modulus:complete representative range1..2;all selected coprime representatives=[1]

phi(2)=1;representatives_1_through_n

right modulus representative1:original nonnegative Euclidean inputs=[1, 7]

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

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

coprime indicator=1;include representative=是

right modulus representative2:original nonnegative Euclidean inputs=[2, 7]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 7, 0, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

right modulus:original representative k=2 in1..7;verified gcd(k,n)=1

coprime indicator=1;include representative=是

right modulus representative3:original nonnegative Euclidean inputs=[3, 7]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 7, 0, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=3 in1..7;verified gcd(k,n)=1

coprime indicator=1;include representative=是

right modulus representative4:original nonnegative Euclidean inputs=[4, 7]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 7, 0, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

right modulus:original representative k=4 in1..7;verified gcd(k,n)=1

coprime indicator=1;include representative=是

right modulus representative5:original nonnegative Euclidean inputs=[5, 7]

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

right modulus:original representative k=5 in1..7;verified gcd(k,n)=1

coprime indicator=1;include representative=是

right modulus representative6:original nonnegative Euclidean inputs=[6, 7]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 7, 0, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal positive gcd=1

right modulus:original representative k=6 in1..7;verified gcd(k,n)=1

coprime indicator=1;include representative=是

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

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

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

coprime indicator=0;include representative=否

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

right modulus exact coprime indicator count = 6

right modulus:complete representative range1..7;all selected coprime representatives=[1, 2, 3, 4, 5, 6]

phi(7)=6;representatives_1_through_n

product modulus representative1:original nonnegative Euclidean inputs=[1, 14]

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

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

coprime indicator=1;include representative=是

product modulus representative2:original nonnegative Euclidean inputs=[2, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 14, 0, 2], [14, 2, 7, 0]];terminal positive gcd=2

product modulus:original representative k=2 in1..14;verified gcd(k,n)=2

coprime indicator=0;include representative=否

product modulus representative3:original nonnegative Euclidean inputs=[3, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 14, 0, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=3 in1..14;verified gcd(k,n)=1

coprime indicator=1;include representative=是

product modulus representative4:original nonnegative Euclidean inputs=[4, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 14, 0, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=4 in1..14;verified gcd(k,n)=2

coprime indicator=0;include representative=否

product modulus representative5:original nonnegative Euclidean inputs=[5, 14]

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

product modulus:original representative k=5 in1..14;verified gcd(k,n)=1

coprime indicator=1;include representative=是

product modulus representative6:original nonnegative Euclidean inputs=[6, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 14, 0, 6], [14, 6, 2, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=6 in1..14;verified gcd(k,n)=2

coprime indicator=0;include representative=否

product modulus representative7:original nonnegative Euclidean inputs=[7, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 14, 0, 7], [14, 7, 2, 0]];terminal positive gcd=7

product modulus:original representative k=7 in1..14;verified gcd(k,n)=7

coprime indicator=0;include representative=否

product modulus representative8:original nonnegative Euclidean inputs=[8, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 14, 0, 8], [14, 8, 1, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

product modulus:original representative k=8 in1..14;verified gcd(k,n)=2

coprime indicator=0;include representative=否

product modulus representative9:original nonnegative Euclidean inputs=[9, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 14, 0, 9], [14, 9, 1, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

product modulus:original representative k=9 in1..14;verified gcd(k,n)=1

coprime indicator=1;include representative=是

product modulus representative10:original nonnegative Euclidean inputs=[10, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 14, 0, 10], [14, 10, 1, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

product modulus:original representative k=10 in1..14;verified gcd(k,n)=2

coprime indicator=0;include representative=否

product modulus representative11:original nonnegative Euclidean inputs=[11, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 14, 0, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

product modulus:original representative k=11 in1..14;verified gcd(k,n)=1

coprime indicator=1;include representative=是

product modulus representative12:original nonnegative Euclidean inputs=[12, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 14, 0, 12], [14, 12, 1, 2], [12, 2, 6, 0]];terminal positive gcd=2

product modulus:original representative k=12 in1..14;verified gcd(k,n)=2

coprime indicator=0;include representative=否

product modulus representative13:original nonnegative Euclidean inputs=[13, 14]

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 14, 0, 13], [14, 13, 1, 1], [13, 1, 13, 0]];terminal positive gcd=1

product modulus:original representative k=13 in1..14;verified gcd(k,n)=1

coprime indicator=1;include representative=是

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

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

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

coprime indicator=0;include representative=否

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

product modulus exact coprime indicator count = 6

product modulus:complete representative range1..14;all selected coprime representatives=[1, 3, 5, 9, 11, 13]

phi(14)=6;representatives_1_through_n

totient supplied instance product of counts := mul(1, 6)

totient supplied instance product of counts = 6

original coprime inputs=[2, 7];gcd=1;product modulus=14;exact totients=[1, 6, 6]

phi(m)*phi(n)=6;same supplied-instance equality=是;instance_only_with_coprimality

逐步反向核對

• S056 · 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。
• S055 · premise_and_result_audit — 以精確有理數重算 totient supplied instance product of counts，不以小數近似替代。
• S054 · 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。
• S053 · premise_and_result_audit — 以精確有理數重算 product modulus exact coprime indicator count，不以小數近似替代。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 right modulus exact coprime indicator count，不以小數近似替代。
• 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 — 以精確有理數重算 left modulus exact coprime indicator count，不以小數近似替代。
• 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 — 以精確有理數重算 totient multiplicativity original modulus product，不以小數近似替代。
• 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

(2, 7, 1, 6, 6, 6, 是, instance_only_with_coprimality)

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。