第 20085 題：Euler定理

原題與已知條件

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。

• G001 — 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。

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-theorem-number-theory

逐步推導

source operation=eulerthm-unit-permutation;original snapshot and mathematics retained

explicit ordered source output fields=[units, sorted_mapped_units, permutation, certificate]

Euler ORIGINAL signed base normalization exact quotient := div(11, 14)

Euler ORIGINAL signed base normalization exact quotient = 11/14

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 14)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(11, 0)

Euler ORIGINAL signed base normalization normalized remainder = 11

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=11;modulus=14;exact ratio=11/14;floor quotient=0

value=q*n+remainder;normalized residue=11 in[0,14);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[11, 14];nonnegative initial pair=[11, 14]

ALL exact Euclidean rows[left,right,q,r]=[[11, 14, 0, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 14];nonnegative initial pair=[0, 14]

ALL exact Euclidean rows[left,right,q,r]=[[0, 14, 0, 0]];terminal gcd=14

Euler residue representative1:ORIGINAL gcd inputs=[1, 14];nonnegative initial pair=[1, 14]

ALL exact Euclidean rows[left,right,q,r]=[[1, 14, 0, 1], [14, 1, 14, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 14];nonnegative initial pair=[2, 14]

ALL exact Euclidean rows[left,right,q,r]=[[2, 14, 0, 2], [14, 2, 7, 0]];terminal gcd=2

Euler residue representative3:ORIGINAL gcd inputs=[3, 14];nonnegative initial pair=[3, 14]

ALL exact Euclidean rows[left,right,q,r]=[[3, 14, 0, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 14];nonnegative initial pair=[4, 14]

ALL exact Euclidean rows[left,right,q,r]=[[4, 14, 0, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative5:ORIGINAL gcd inputs=[5, 14];nonnegative initial pair=[5, 14]

ALL exact Euclidean rows[left,right,q,r]=[[5, 14, 0, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 14];nonnegative initial pair=[6, 14]

ALL exact Euclidean rows[left,right,q,r]=[[6, 14, 0, 6], [14, 6, 2, 2], [6, 2, 3, 0]];terminal gcd=2

Euler residue representative7:ORIGINAL gcd inputs=[7, 14];nonnegative initial pair=[7, 14]

ALL exact Euclidean rows[left,right,q,r]=[[7, 14, 0, 7], [14, 7, 2, 0]];terminal gcd=7

Euler residue representative8:ORIGINAL gcd inputs=[8, 14];nonnegative initial pair=[8, 14]

ALL exact Euclidean rows[left,right,q,r]=[[8, 14, 0, 8], [14, 8, 1, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal gcd=2

Euler residue representative9:ORIGINAL gcd inputs=[9, 14];nonnegative initial pair=[9, 14]

ALL exact Euclidean rows[left,right,q,r]=[[9, 14, 0, 9], [14, 9, 1, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 14];nonnegative initial pair=[10, 14]

ALL exact Euclidean rows[left,right,q,r]=[[10, 14, 0, 10], [14, 10, 1, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative11:ORIGINAL gcd inputs=[11, 14];nonnegative initial pair=[11, 14]

ALL exact Euclidean rows[left,right,q,r]=[[11, 14, 0, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 14];nonnegative initial pair=[12, 14]

ALL exact Euclidean rows[left,right,q,r]=[[12, 14, 0, 12], [14, 12, 1, 2], [12, 2, 6, 0]];terminal gcd=2

Euler residue representative13:ORIGINAL gcd inputs=[13, 14];nonnegative initial pair=[13, 14]

ALL exact Euclidean rows[left,right,q,r]=[[13, 14, 0, 13], [14, 13, 1, 1], [13, 1, 13, 0]];terminal gcd=1

Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1)

Euler complete reduced residue count phi = 6

COMPLETE original residue carrier0..13;ALL [representative,gcd,unit] rows=[[0, 14, 否], [1, 1, 是], [2, 2, 否], [3, 1, 是], [4, 2, 否], [5, 1, 是], [6, 2, 否], [7, 7, 否], [8, 2, 否], [9, 1, 是], [10, 2, 否], [11, 1, 是], [12, 2, 否], [13, 1, 是]]

increasing COMPLETE U(n)=[1, 3, 5, 9, 11, 13];phi(n)=6;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(11, 9)

Euler ACTUAL base-unit inverse product = 99

Euler ACTUAL inverse check exact quotient := div(99, 14)

Euler ACTUAL inverse check exact quotient = 99/14

Euler ACTUAL inverse check floor quotient multiple := mul(7, 14)

Euler ACTUAL inverse check floor quotient multiple = 98

Euler ACTUAL inverse check normalized remainder := sub(99, 98)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=99;modulus=14;exact ratio=99/14;floor quotient=7

value=q*n+remainder;normalized residue=1 in[0,14);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=14;gcd(original a=11,n)=1;normalized base=11;ACTUAL inverse=9;COMPLETE U(n)=[1, 3, 5, 9, 11, 13]

for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)

Euler ORIGINAL a times unit representative1 := mul(11, 1)

Euler ORIGINAL a times unit representative1 = 11

Euler unit representative1 raw image exact quotient := div(11, 14)

Euler unit representative1 raw image exact quotient = 11/14

Euler unit representative1 raw image floor quotient multiple := mul(0, 14)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(11, 0)

Euler unit representative1 raw image normalized remainder = 11

Euler unit representative1 raw image:ORIGINAL signed integer=11;modulus=14;exact ratio=11/14;floor quotient=0

value=q*n+remainder;normalized residue=11 in[0,14);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(11, 3)

Euler ORIGINAL a times unit representative3 = 33

Euler unit representative3 raw image exact quotient := div(33, 14)

Euler unit representative3 raw image exact quotient = 33/14

Euler unit representative3 raw image floor quotient multiple := mul(2, 14)

Euler unit representative3 raw image floor quotient multiple = 28

Euler unit representative3 raw image normalized remainder := sub(33, 28)

Euler unit representative3 raw image normalized remainder = 5

Euler unit representative3 raw image:ORIGINAL signed integer=33;modulus=14;exact ratio=33/14;floor quotient=2

value=q*n+remainder;normalized residue=5 in[0,14);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(11, 5)

Euler ORIGINAL a times unit representative5 = 55

Euler unit representative5 raw image exact quotient := div(55, 14)

Euler unit representative5 raw image exact quotient = 55/14

Euler unit representative5 raw image floor quotient multiple := mul(3, 14)

Euler unit representative5 raw image floor quotient multiple = 42

Euler unit representative5 raw image normalized remainder := sub(55, 42)

Euler unit representative5 raw image normalized remainder = 13

Euler unit representative5 raw image:ORIGINAL signed integer=55;modulus=14;exact ratio=55/14;floor quotient=3

value=q*n+remainder;normalized residue=13 in[0,14);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(11, 9)

Euler ORIGINAL a times unit representative9 = 99

Euler unit representative9 raw image exact quotient := div(99, 14)

Euler unit representative9 raw image exact quotient = 99/14

Euler unit representative9 raw image floor quotient multiple := mul(7, 14)

Euler unit representative9 raw image floor quotient multiple = 98

Euler unit representative9 raw image normalized remainder := sub(99, 98)

Euler unit representative9 raw image normalized remainder = 1

Euler unit representative9 raw image:ORIGINAL signed integer=99;modulus=14;exact ratio=99/14;floor quotient=7

value=q*n+remainder;normalized residue=1 in[0,14);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(11, 11)

Euler ORIGINAL a times unit representative11 = 121

Euler unit representative11 raw image exact quotient := div(121, 14)

Euler unit representative11 raw image exact quotient = 121/14

Euler unit representative11 raw image floor quotient multiple := mul(8, 14)

Euler unit representative11 raw image floor quotient multiple = 112

Euler unit representative11 raw image normalized remainder := sub(121, 112)

Euler unit representative11 raw image normalized remainder = 9

Euler unit representative11 raw image:ORIGINAL signed integer=121;modulus=14;exact ratio=121/14;floor quotient=8

value=q*n+remainder;normalized residue=9 in[0,14);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative13 := mul(11, 13)

Euler ORIGINAL a times unit representative13 = 143

Euler unit representative13 raw image exact quotient := div(143, 14)

Euler unit representative13 raw image exact quotient = 143/14

Euler unit representative13 raw image floor quotient multiple := mul(10, 14)

Euler unit representative13 raw image floor quotient multiple = 140

Euler unit representative13 raw image normalized remainder := sub(143, 140)

Euler unit representative13 raw image normalized remainder = 3

Euler unit representative13 raw image:ORIGINAL signed integer=143;modulus=14;exact ratio=143/14;floor quotient=10

value=q*n+remainder;normalized residue=3 in[0,14);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 3, 5, 9, 11, 13];actual images in ORIGINAL unit order=[11, 5, 13, 1, 9, 3]

SOURCE output sorts the mapped list numerically=[1, 3, 5, 9, 11, 13];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S059 · 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。
• S058 · 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。
• S057 · premise_and_result_audit — 以精確有理數重算 Euler unit representative13 raw image normalized remainder，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Euler unit representative13 raw image floor quotient multiple，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Euler unit representative13 raw image exact quotient，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative13，不以小數近似替代。
• S053 · 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。
• S052 · premise_and_result_audit — 以精確有理數重算 Euler unit representative11 raw image normalized remainder，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Euler unit representative11 raw image floor quotient multiple，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Euler unit representative11 raw image exact quotient，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative11，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative9 raw image normalized remainder，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Euler unit representative9 raw image floor quotient multiple，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Euler unit representative9 raw image exact quotient，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative9，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative5 raw image normalized remainder，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image floor quotient multiple，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image exact quotient，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative5，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 — 以精確有理數重算 Euler unit representative3 raw image normalized remainder，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image floor quotient multiple，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image exact quotient，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative3，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image floor quotient multiple，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image exact quotient，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative1，不以小數近似替代。
• 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 — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check floor quotient multiple，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check exact quotient，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL base-unit inverse product，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 — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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 — 以精確有理數重算 Euler ORIGINAL signed base normalization normalized remainder，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization floor quotient multiple，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization exact quotient，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

([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], 是, multiplication_by_a_permutes_reduced_residues)

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。