第 20084 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":13,"a":10}。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":13,"a":10}。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(10, 13)

Euler ORIGINAL signed base normalization exact quotient = 10/13

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 13)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(10, 0)

Euler ORIGINAL signed base normalization normalized remainder = 10

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=10;modulus=13;exact ratio=10/13;floor quotient=0

value=q*n+remainder;normalized residue=10 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[10, 13];nonnegative initial pair=[10, 13]

ALL exact Euclidean rows[left,right,q,r]=[[10, 13, 0, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 13];nonnegative initial pair=[0, 13]

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

Euler residue representative1:ORIGINAL gcd inputs=[1, 13];nonnegative initial pair=[1, 13]

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

Euler residue representative2:ORIGINAL gcd inputs=[2, 13];nonnegative initial pair=[2, 13]

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

Euler residue representative3:ORIGINAL gcd inputs=[3, 13];nonnegative initial pair=[3, 13]

ALL exact Euclidean rows[left,right,q,r]=[[3, 13, 0, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 13];nonnegative initial pair=[4, 13]

ALL exact Euclidean rows[left,right,q,r]=[[4, 13, 0, 4], [13, 4, 3, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 13];nonnegative initial pair=[5, 13]

ALL exact Euclidean rows[left,right,q,r]=[[5, 13, 0, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 13];nonnegative initial pair=[6, 13]

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

Euler residue representative7:ORIGINAL gcd inputs=[7, 13];nonnegative initial pair=[7, 13]

ALL exact Euclidean rows[left,right,q,r]=[[7, 13, 0, 7], [13, 7, 1, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 13];nonnegative initial pair=[8, 13]

ALL exact Euclidean rows[left,right,q,r]=[[8, 13, 0, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 13];nonnegative initial pair=[9, 13]

ALL exact Euclidean rows[left,right,q,r]=[[9, 13, 0, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 13];nonnegative initial pair=[10, 13]

ALL exact Euclidean rows[left,right,q,r]=[[10, 13, 0, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 13];nonnegative initial pair=[11, 13]

ALL exact Euclidean rows[left,right,q,r]=[[11, 13, 0, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 13];nonnegative initial pair=[12, 13]

ALL exact Euclidean rows[left,right,q,r]=[[12, 13, 0, 12], [13, 12, 1, 1], [12, 1, 12, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 12

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

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];phi(n)=12;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(10, 4)

Euler ACTUAL base-unit inverse product = 40

Euler ACTUAL inverse check exact quotient := div(40, 13)

Euler ACTUAL inverse check exact quotient = 40/13

Euler ACTUAL inverse check floor quotient multiple := mul(3, 13)

Euler ACTUAL inverse check floor quotient multiple = 39

Euler ACTUAL inverse check normalized remainder := sub(40, 39)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=40;modulus=13;exact ratio=40/13;floor quotient=3

value=q*n+remainder;normalized residue=1 in[0,13);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=13;gcd(original a=10,n)=1;normalized base=10;ACTUAL inverse=4;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

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(10, 1)

Euler ORIGINAL a times unit representative1 = 10

Euler unit representative1 raw image exact quotient := div(10, 13)

Euler unit representative1 raw image exact quotient = 10/13

Euler unit representative1 raw image floor quotient multiple := mul(0, 13)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(10, 0)

Euler unit representative1 raw image normalized remainder = 10

Euler unit representative1 raw image:ORIGINAL signed integer=10;modulus=13;exact ratio=10/13;floor quotient=0

value=q*n+remainder;normalized residue=10 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(10, 2)

Euler ORIGINAL a times unit representative2 = 20

Euler unit representative2 raw image exact quotient := div(20, 13)

Euler unit representative2 raw image exact quotient = 20/13

Euler unit representative2 raw image floor quotient multiple := mul(1, 13)

Euler unit representative2 raw image floor quotient multiple = 13

Euler unit representative2 raw image normalized remainder := sub(20, 13)

Euler unit representative2 raw image normalized remainder = 7

Euler unit representative2 raw image:ORIGINAL signed integer=20;modulus=13;exact ratio=20/13;floor quotient=1

value=q*n+remainder;normalized residue=7 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(10, 3)

Euler ORIGINAL a times unit representative3 = 30

Euler unit representative3 raw image exact quotient := div(30, 13)

Euler unit representative3 raw image exact quotient = 30/13

Euler unit representative3 raw image floor quotient multiple := mul(2, 13)

Euler unit representative3 raw image floor quotient multiple = 26

Euler unit representative3 raw image normalized remainder := sub(30, 26)

Euler unit representative3 raw image normalized remainder = 4

Euler unit representative3 raw image:ORIGINAL signed integer=30;modulus=13;exact ratio=30/13;floor quotient=2

value=q*n+remainder;normalized residue=4 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(10, 4)

Euler ORIGINAL a times unit representative4 = 40

Euler unit representative4 raw image exact quotient := div(40, 13)

Euler unit representative4 raw image exact quotient = 40/13

Euler unit representative4 raw image floor quotient multiple := mul(3, 13)

Euler unit representative4 raw image floor quotient multiple = 39

Euler unit representative4 raw image normalized remainder := sub(40, 39)

Euler unit representative4 raw image normalized remainder = 1

Euler unit representative4 raw image:ORIGINAL signed integer=40;modulus=13;exact ratio=40/13;floor quotient=3

value=q*n+remainder;normalized residue=1 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(10, 5)

Euler ORIGINAL a times unit representative5 = 50

Euler unit representative5 raw image exact quotient := div(50, 13)

Euler unit representative5 raw image exact quotient = 50/13

Euler unit representative5 raw image floor quotient multiple := mul(3, 13)

Euler unit representative5 raw image floor quotient multiple = 39

Euler unit representative5 raw image normalized remainder := sub(50, 39)

Euler unit representative5 raw image normalized remainder = 11

Euler unit representative5 raw image:ORIGINAL signed integer=50;modulus=13;exact ratio=50/13;floor quotient=3

value=q*n+remainder;normalized residue=11 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative6 := mul(10, 6)

Euler ORIGINAL a times unit representative6 = 60

Euler unit representative6 raw image exact quotient := div(60, 13)

Euler unit representative6 raw image exact quotient = 60/13

Euler unit representative6 raw image floor quotient multiple := mul(4, 13)

Euler unit representative6 raw image floor quotient multiple = 52

Euler unit representative6 raw image normalized remainder := sub(60, 52)

Euler unit representative6 raw image normalized remainder = 8

Euler unit representative6 raw image:ORIGINAL signed integer=60;modulus=13;exact ratio=60/13;floor quotient=4

value=q*n+remainder;normalized residue=8 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(10, 7)

Euler ORIGINAL a times unit representative7 = 70

Euler unit representative7 raw image exact quotient := div(70, 13)

Euler unit representative7 raw image exact quotient = 70/13

Euler unit representative7 raw image floor quotient multiple := mul(5, 13)

Euler unit representative7 raw image floor quotient multiple = 65

Euler unit representative7 raw image normalized remainder := sub(70, 65)

Euler unit representative7 raw image normalized remainder = 5

Euler unit representative7 raw image:ORIGINAL signed integer=70;modulus=13;exact ratio=70/13;floor quotient=5

value=q*n+remainder;normalized residue=5 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(10, 8)

Euler ORIGINAL a times unit representative8 = 80

Euler unit representative8 raw image exact quotient := div(80, 13)

Euler unit representative8 raw image exact quotient = 80/13

Euler unit representative8 raw image floor quotient multiple := mul(6, 13)

Euler unit representative8 raw image floor quotient multiple = 78

Euler unit representative8 raw image normalized remainder := sub(80, 78)

Euler unit representative8 raw image normalized remainder = 2

Euler unit representative8 raw image:ORIGINAL signed integer=80;modulus=13;exact ratio=80/13;floor quotient=6

value=q*n+remainder;normalized residue=2 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(10, 9)

Euler ORIGINAL a times unit representative9 = 90

Euler unit representative9 raw image exact quotient := div(90, 13)

Euler unit representative9 raw image exact quotient = 90/13

Euler unit representative9 raw image floor quotient multiple := mul(6, 13)

Euler unit representative9 raw image floor quotient multiple = 78

Euler unit representative9 raw image normalized remainder := sub(90, 78)

Euler unit representative9 raw image normalized remainder = 12

Euler unit representative9 raw image:ORIGINAL signed integer=90;modulus=13;exact ratio=90/13;floor quotient=6

value=q*n+remainder;normalized residue=12 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative10 := mul(10, 10)

Euler ORIGINAL a times unit representative10 = 100

Euler unit representative10 raw image exact quotient := div(100, 13)

Euler unit representative10 raw image exact quotient = 100/13

Euler unit representative10 raw image floor quotient multiple := mul(7, 13)

Euler unit representative10 raw image floor quotient multiple = 91

Euler unit representative10 raw image normalized remainder := sub(100, 91)

Euler unit representative10 raw image normalized remainder = 9

Euler unit representative10 raw image:ORIGINAL signed integer=100;modulus=13;exact ratio=100/13;floor quotient=7

value=q*n+remainder;normalized residue=9 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(10, 11)

Euler ORIGINAL a times unit representative11 = 110

Euler unit representative11 raw image exact quotient := div(110, 13)

Euler unit representative11 raw image exact quotient = 110/13

Euler unit representative11 raw image floor quotient multiple := mul(8, 13)

Euler unit representative11 raw image floor quotient multiple = 104

Euler unit representative11 raw image normalized remainder := sub(110, 104)

Euler unit representative11 raw image normalized remainder = 6

Euler unit representative11 raw image:ORIGINAL signed integer=110;modulus=13;exact ratio=110/13;floor quotient=8

value=q*n+remainder;normalized residue=6 in[0,13);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative12 := mul(10, 12)

Euler ORIGINAL a times unit representative12 = 120

Euler unit representative12 raw image exact quotient := div(120, 13)

Euler unit representative12 raw image exact quotient = 120/13

Euler unit representative12 raw image floor quotient multiple := mul(9, 13)

Euler unit representative12 raw image floor quotient multiple = 117

Euler unit representative12 raw image normalized remainder := sub(120, 117)

Euler unit representative12 raw image normalized remainder = 3

Euler unit representative12 raw image:ORIGINAL signed integer=120;modulus=13;exact ratio=120/13;floor quotient=9

value=q*n+remainder;normalized residue=3 in[0,13);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];actual images in ORIGINAL unit order=[10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3]

SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S088 · 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。
• S087 · 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。
• S086 · premise_and_result_audit — 以精確有理數重算 Euler unit representative12 raw image normalized remainder，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Euler unit representative12 raw image floor quotient multiple，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Euler unit representative12 raw image exact quotient，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative12，不以小數近似替代。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · premise_and_result_audit — 以精確有理數重算 Euler unit representative10 raw image normalized remainder，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Euler unit representative10 raw image floor quotient multiple，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Euler unit representative10 raw image exact quotient，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative10，不以小數近似替代。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · premise_and_result_audit — 以精確有理數重算 Euler unit representative6 raw image normalized remainder，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Euler unit representative6 raw image floor quotient multiple，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Euler unit representative6 raw image exact quotient，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative6，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 — 以精確有理數重算 Euler unit representative5 raw image normalized remainder，不以小數近似替代。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• 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 — 以精確有理數重算 Euler unit representative4 raw image normalized remainder，不以小數近似替代。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• 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 — 以精確有理數重算 Euler unit representative3 raw image normalized remainder，不以小數近似替代。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• 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 — 以精確有理數重算 Euler unit representative2 raw image normalized remainder，不以小數近似替代。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• 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 — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，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 floor quotient multiple，不以小數近似替代。
• S029 · 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。
• S028 · 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。
• 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 — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，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 floor quotient multiple，不以小數近似替代。
• S023 · 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。
• S022 · 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。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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, 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。