第 20086 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":15,"a":13}。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":15,"a":13}。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(13, 15)

Euler ORIGINAL signed base normalization exact quotient = 13/15

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 15)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(13, 0)

Euler ORIGINAL signed base normalization normalized remainder = 13

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=13;modulus=15;exact ratio=13/15;floor quotient=0

value=q*n+remainder;normalized residue=13 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[13, 15];nonnegative initial pair=[13, 15]

ALL exact Euclidean rows[left,right,q,r]=[[13, 15, 0, 13], [15, 13, 1, 2], [13, 2, 6, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 15];nonnegative initial pair=[0, 15]

ALL exact Euclidean rows[left,right,q,r]=[[0, 15, 0, 0]];terminal gcd=15

Euler residue representative1:ORIGINAL gcd inputs=[1, 15];nonnegative initial pair=[1, 15]

ALL exact Euclidean rows[left,right,q,r]=[[1, 15, 0, 1], [15, 1, 15, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 15];nonnegative initial pair=[2, 15]

ALL exact Euclidean rows[left,right,q,r]=[[2, 15, 0, 2], [15, 2, 7, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 15];nonnegative initial pair=[3, 15]

ALL exact Euclidean rows[left,right,q,r]=[[3, 15, 0, 3], [15, 3, 5, 0]];terminal gcd=3

Euler residue representative4:ORIGINAL gcd inputs=[4, 15];nonnegative initial pair=[4, 15]

ALL exact Euclidean rows[left,right,q,r]=[[4, 15, 0, 4], [15, 4, 3, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 15];nonnegative initial pair=[5, 15]

ALL exact Euclidean rows[left,right,q,r]=[[5, 15, 0, 5], [15, 5, 3, 0]];terminal gcd=5

Euler residue representative6:ORIGINAL gcd inputs=[6, 15];nonnegative initial pair=[6, 15]

ALL exact Euclidean rows[left,right,q,r]=[[6, 15, 0, 6], [15, 6, 2, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative7:ORIGINAL gcd inputs=[7, 15];nonnegative initial pair=[7, 15]

ALL exact Euclidean rows[left,right,q,r]=[[7, 15, 0, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 15];nonnegative initial pair=[8, 15]

ALL exact Euclidean rows[left,right,q,r]=[[8, 15, 0, 8], [15, 8, 1, 7], [8, 7, 1, 1], [7, 1, 7, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 15];nonnegative initial pair=[9, 15]

ALL exact Euclidean rows[left,right,q,r]=[[9, 15, 0, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative10:ORIGINAL gcd inputs=[10, 15];nonnegative initial pair=[10, 15]

ALL exact Euclidean rows[left,right,q,r]=[[10, 15, 0, 10], [15, 10, 1, 5], [10, 5, 2, 0]];terminal gcd=5

Euler residue representative11:ORIGINAL gcd inputs=[11, 15];nonnegative initial pair=[11, 15]

ALL exact Euclidean rows[left,right,q,r]=[[11, 15, 0, 11], [15, 11, 1, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 15];nonnegative initial pair=[12, 15]

ALL exact Euclidean rows[left,right,q,r]=[[12, 15, 0, 12], [15, 12, 1, 3], [12, 3, 4, 0]];terminal gcd=3

Euler residue representative13:ORIGINAL gcd inputs=[13, 15];nonnegative initial pair=[13, 15]

ALL exact Euclidean rows[left,right,q,r]=[[13, 15, 0, 13], [15, 13, 1, 2], [13, 2, 6, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 15];nonnegative initial pair=[14, 15]

ALL exact Euclidean rows[left,right,q,r]=[[14, 15, 0, 14], [15, 14, 1, 1], [14, 1, 14, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 8

COMPLETE original residue carrier0..14;ALL [representative,gcd,unit] rows=[[0, 15, 否], [1, 1, 是], [2, 1, 是], [3, 3, 否], [4, 1, 是], [5, 5, 否], [6, 3, 否], [7, 1, 是], [8, 1, 是], [9, 3, 否], [10, 5, 否], [11, 1, 是], [12, 3, 否], [13, 1, 是], [14, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 4, 7, 8, 11, 13, 14];phi(n)=8;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(13, 7)

Euler ACTUAL base-unit inverse product = 91

Euler ACTUAL inverse check exact quotient := div(91, 15)

Euler ACTUAL inverse check exact quotient = 91/15

Euler ACTUAL inverse check floor quotient multiple := mul(6, 15)

Euler ACTUAL inverse check floor quotient multiple = 90

Euler ACTUAL inverse check normalized remainder := sub(91, 90)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=91;modulus=15;exact ratio=91/15;floor quotient=6

value=q*n+remainder;normalized residue=1 in[0,15);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=15;gcd(original a=13,n)=1;normalized base=13;ACTUAL inverse=7;COMPLETE U(n)=[1, 2, 4, 7, 8, 11, 13, 14]

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(13, 1)

Euler ORIGINAL a times unit representative1 = 13

Euler unit representative1 raw image exact quotient := div(13, 15)

Euler unit representative1 raw image exact quotient = 13/15

Euler unit representative1 raw image floor quotient multiple := mul(0, 15)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(13, 0)

Euler unit representative1 raw image normalized remainder = 13

Euler unit representative1 raw image:ORIGINAL signed integer=13;modulus=15;exact ratio=13/15;floor quotient=0

value=q*n+remainder;normalized residue=13 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(13, 2)

Euler ORIGINAL a times unit representative2 = 26

Euler unit representative2 raw image exact quotient := div(26, 15)

Euler unit representative2 raw image exact quotient = 26/15

Euler unit representative2 raw image floor quotient multiple := mul(1, 15)

Euler unit representative2 raw image floor quotient multiple = 15

Euler unit representative2 raw image normalized remainder := sub(26, 15)

Euler unit representative2 raw image normalized remainder = 11

Euler unit representative2 raw image:ORIGINAL signed integer=26;modulus=15;exact ratio=26/15;floor quotient=1

value=q*n+remainder;normalized residue=11 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(13, 4)

Euler ORIGINAL a times unit representative4 = 52

Euler unit representative4 raw image exact quotient := div(52, 15)

Euler unit representative4 raw image exact quotient = 52/15

Euler unit representative4 raw image floor quotient multiple := mul(3, 15)

Euler unit representative4 raw image floor quotient multiple = 45

Euler unit representative4 raw image normalized remainder := sub(52, 45)

Euler unit representative4 raw image normalized remainder = 7

Euler unit representative4 raw image:ORIGINAL signed integer=52;modulus=15;exact ratio=52/15;floor quotient=3

value=q*n+remainder;normalized residue=7 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(13, 7)

Euler ORIGINAL a times unit representative7 = 91

Euler unit representative7 raw image exact quotient := div(91, 15)

Euler unit representative7 raw image exact quotient = 91/15

Euler unit representative7 raw image floor quotient multiple := mul(6, 15)

Euler unit representative7 raw image floor quotient multiple = 90

Euler unit representative7 raw image normalized remainder := sub(91, 90)

Euler unit representative7 raw image normalized remainder = 1

Euler unit representative7 raw image:ORIGINAL signed integer=91;modulus=15;exact ratio=91/15;floor quotient=6

value=q*n+remainder;normalized residue=1 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(13, 8)

Euler ORIGINAL a times unit representative8 = 104

Euler unit representative8 raw image exact quotient := div(104, 15)

Euler unit representative8 raw image exact quotient = 104/15

Euler unit representative8 raw image floor quotient multiple := mul(6, 15)

Euler unit representative8 raw image floor quotient multiple = 90

Euler unit representative8 raw image normalized remainder := sub(104, 90)

Euler unit representative8 raw image normalized remainder = 14

Euler unit representative8 raw image:ORIGINAL signed integer=104;modulus=15;exact ratio=104/15;floor quotient=6

value=q*n+remainder;normalized residue=14 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(13, 11)

Euler ORIGINAL a times unit representative11 = 143

Euler unit representative11 raw image exact quotient := div(143, 15)

Euler unit representative11 raw image exact quotient = 143/15

Euler unit representative11 raw image floor quotient multiple := mul(9, 15)

Euler unit representative11 raw image floor quotient multiple = 135

Euler unit representative11 raw image normalized remainder := sub(143, 135)

Euler unit representative11 raw image normalized remainder = 8

Euler unit representative11 raw image:ORIGINAL signed integer=143;modulus=15;exact ratio=143/15;floor quotient=9

value=q*n+remainder;normalized residue=8 in[0,15);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative13 = 169

Euler unit representative13 raw image exact quotient := div(169, 15)

Euler unit representative13 raw image exact quotient = 169/15

Euler unit representative13 raw image floor quotient multiple := mul(11, 15)

Euler unit representative13 raw image floor quotient multiple = 165

Euler unit representative13 raw image normalized remainder := sub(169, 165)

Euler unit representative13 raw image normalized remainder = 4

Euler unit representative13 raw image:ORIGINAL signed integer=169;modulus=15;exact ratio=169/15;floor quotient=11

value=q*n+remainder;normalized residue=4 in[0,15);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative14 := mul(13, 14)

Euler ORIGINAL a times unit representative14 = 182

Euler unit representative14 raw image exact quotient := div(182, 15)

Euler unit representative14 raw image exact quotient = 182/15

Euler unit representative14 raw image floor quotient multiple := mul(12, 15)

Euler unit representative14 raw image floor quotient multiple = 180

Euler unit representative14 raw image normalized remainder := sub(182, 180)

Euler unit representative14 raw image normalized remainder = 2

Euler unit representative14 raw image:ORIGINAL signed integer=182;modulus=15;exact ratio=182/15;floor quotient=12

value=q*n+remainder;normalized residue=2 in[0,15);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 2, 4, 7, 8, 11, 13, 14];actual images in ORIGINAL unit order=[13, 11, 7, 1, 14, 8, 4, 2]

SOURCE output sorts the mapped list numerically=[1, 2, 4, 7, 8, 11, 13, 14];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S070 · 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。
• S069 · 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。
• S068 · premise_and_result_audit — 以精確有理數重算 Euler unit representative14 raw image normalized remainder，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Euler unit representative14 raw image floor quotient multiple，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Euler unit representative14 raw image exact quotient，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative14，不以小數近似替代。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• 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 — 以精確有理數重算 Euler unit representative11 raw image normalized remainder，不以小數近似替代。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• 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 Euler unit representative8 raw image normalized remainder，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Euler unit representative8 raw image floor quotient multiple，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Euler unit representative8 raw image exact quotient，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative8，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative7 raw image normalized remainder，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Euler unit representative7 raw image floor quotient multiple，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Euler unit representative7 raw image exact quotient，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative7，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative4 raw image normalized remainder，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Euler unit representative4 raw image floor quotient multiple，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Euler unit representative4 raw image exact quotient，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative4，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative2 raw image normalized remainder，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Euler unit representative2 raw image floor quotient multiple，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Euler unit representative2 raw image exact quotient，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative2，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，不以小數近似替代。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• 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 — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S026 · 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。
• S025 · 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。
• S024 · 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。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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 — 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, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], 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。