第 20088 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":17,"a":14}。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":17,"a":14}。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(14, 17)

Euler ORIGINAL signed base normalization exact quotient = 14/17

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 17)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(14, 0)

Euler ORIGINAL signed base normalization normalized remainder = 14

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=14;modulus=17;exact ratio=14/17;floor quotient=0

value=q*n+remainder;normalized residue=14 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[14, 17];nonnegative initial pair=[14, 17]

ALL exact Euclidean rows[left,right,q,r]=[[14, 17, 0, 14], [17, 14, 1, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 17];nonnegative initial pair=[0, 17]

ALL exact Euclidean rows[left,right,q,r]=[[0, 17, 0, 0]];terminal gcd=17

Euler residue representative1:ORIGINAL gcd inputs=[1, 17];nonnegative initial pair=[1, 17]

ALL exact Euclidean rows[left,right,q,r]=[[1, 17, 0, 1], [17, 1, 17, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 17];nonnegative initial pair=[2, 17]

ALL exact Euclidean rows[left,right,q,r]=[[2, 17, 0, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 17];nonnegative initial pair=[3, 17]

ALL exact Euclidean rows[left,right,q,r]=[[3, 17, 0, 3], [17, 3, 5, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 17];nonnegative initial pair=[4, 17]

ALL exact Euclidean rows[left,right,q,r]=[[4, 17, 0, 4], [17, 4, 4, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 17];nonnegative initial pair=[5, 17]

ALL exact Euclidean rows[left,right,q,r]=[[5, 17, 0, 5], [17, 5, 3, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 17];nonnegative initial pair=[6, 17]

ALL exact Euclidean rows[left,right,q,r]=[[6, 17, 0, 6], [17, 6, 2, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 17];nonnegative initial pair=[7, 17]

ALL exact Euclidean rows[left,right,q,r]=[[7, 17, 0, 7], [17, 7, 2, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 17];nonnegative initial pair=[8, 17]

ALL exact Euclidean rows[left,right,q,r]=[[8, 17, 0, 8], [17, 8, 2, 1], [8, 1, 8, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 17];nonnegative initial pair=[9, 17]

ALL exact Euclidean rows[left,right,q,r]=[[9, 17, 0, 9], [17, 9, 1, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 17];nonnegative initial pair=[10, 17]

ALL exact Euclidean rows[left,right,q,r]=[[10, 17, 0, 10], [17, 10, 1, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 17];nonnegative initial pair=[11, 17]

ALL exact Euclidean rows[left,right,q,r]=[[11, 17, 0, 11], [17, 11, 1, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 17];nonnegative initial pair=[12, 17]

ALL exact Euclidean rows[left,right,q,r]=[[12, 17, 0, 12], [17, 12, 1, 5], [12, 5, 2, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative13:ORIGINAL gcd inputs=[13, 17];nonnegative initial pair=[13, 17]

ALL exact Euclidean rows[left,right,q,r]=[[13, 17, 0, 13], [17, 13, 1, 4], [13, 4, 3, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 17];nonnegative initial pair=[14, 17]

ALL exact Euclidean rows[left,right,q,r]=[[14, 17, 0, 14], [17, 14, 1, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative15:ORIGINAL gcd inputs=[15, 17];nonnegative initial pair=[15, 17]

ALL exact Euclidean rows[left,right,q,r]=[[15, 17, 0, 15], [17, 15, 1, 2], [15, 2, 7, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 17];nonnegative initial pair=[16, 17]

ALL exact Euclidean rows[left,right,q,r]=[[16, 17, 0, 16], [17, 16, 1, 1], [16, 1, 16, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 16

COMPLETE original residue carrier0..16;ALL [representative,gcd,unit] rows=[[0, 17, 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], [13, 1, 是], [14, 1, 是], [15, 1, 是], [16, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];phi(n)=16;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(14, 11)

Euler ACTUAL base-unit inverse product = 154

Euler ACTUAL inverse check exact quotient := div(154, 17)

Euler ACTUAL inverse check exact quotient = 154/17

Euler ACTUAL inverse check floor quotient multiple := mul(9, 17)

Euler ACTUAL inverse check floor quotient multiple = 153

Euler ACTUAL inverse check normalized remainder := sub(154, 153)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=154;modulus=17;exact ratio=154/17;floor quotient=9

value=q*n+remainder;normalized residue=1 in[0,17);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=17;gcd(original a=14,n)=1;normalized base=14;ACTUAL inverse=11;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

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(14, 1)

Euler ORIGINAL a times unit representative1 = 14

Euler unit representative1 raw image exact quotient := div(14, 17)

Euler unit representative1 raw image exact quotient = 14/17

Euler unit representative1 raw image floor quotient multiple := mul(0, 17)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(14, 0)

Euler unit representative1 raw image normalized remainder = 14

Euler unit representative1 raw image:ORIGINAL signed integer=14;modulus=17;exact ratio=14/17;floor quotient=0

value=q*n+remainder;normalized residue=14 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(14, 2)

Euler ORIGINAL a times unit representative2 = 28

Euler unit representative2 raw image exact quotient := div(28, 17)

Euler unit representative2 raw image exact quotient = 28/17

Euler unit representative2 raw image floor quotient multiple := mul(1, 17)

Euler unit representative2 raw image floor quotient multiple = 17

Euler unit representative2 raw image normalized remainder := sub(28, 17)

Euler unit representative2 raw image normalized remainder = 11

Euler unit representative2 raw image:ORIGINAL signed integer=28;modulus=17;exact ratio=28/17;floor quotient=1

value=q*n+remainder;normalized residue=11 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(14, 3)

Euler ORIGINAL a times unit representative3 = 42

Euler unit representative3 raw image exact quotient := div(42, 17)

Euler unit representative3 raw image exact quotient = 42/17

Euler unit representative3 raw image floor quotient multiple := mul(2, 17)

Euler unit representative3 raw image floor quotient multiple = 34

Euler unit representative3 raw image normalized remainder := sub(42, 34)

Euler unit representative3 raw image normalized remainder = 8

Euler unit representative3 raw image:ORIGINAL signed integer=42;modulus=17;exact ratio=42/17;floor quotient=2

value=q*n+remainder;normalized residue=8 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(14, 4)

Euler ORIGINAL a times unit representative4 = 56

Euler unit representative4 raw image exact quotient := div(56, 17)

Euler unit representative4 raw image exact quotient = 56/17

Euler unit representative4 raw image floor quotient multiple := mul(3, 17)

Euler unit representative4 raw image floor quotient multiple = 51

Euler unit representative4 raw image normalized remainder := sub(56, 51)

Euler unit representative4 raw image normalized remainder = 5

Euler unit representative4 raw image:ORIGINAL signed integer=56;modulus=17;exact ratio=56/17;floor quotient=3

value=q*n+remainder;normalized residue=5 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(14, 5)

Euler ORIGINAL a times unit representative5 = 70

Euler unit representative5 raw image exact quotient := div(70, 17)

Euler unit representative5 raw image exact quotient = 70/17

Euler unit representative5 raw image floor quotient multiple := mul(4, 17)

Euler unit representative5 raw image floor quotient multiple = 68

Euler unit representative5 raw image normalized remainder := sub(70, 68)

Euler unit representative5 raw image normalized remainder = 2

Euler unit representative5 raw image:ORIGINAL signed integer=70;modulus=17;exact ratio=70/17;floor quotient=4

value=q*n+remainder;normalized residue=2 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative6 := mul(14, 6)

Euler ORIGINAL a times unit representative6 = 84

Euler unit representative6 raw image exact quotient := div(84, 17)

Euler unit representative6 raw image exact quotient = 84/17

Euler unit representative6 raw image floor quotient multiple := mul(4, 17)

Euler unit representative6 raw image floor quotient multiple = 68

Euler unit representative6 raw image normalized remainder := sub(84, 68)

Euler unit representative6 raw image normalized remainder = 16

Euler unit representative6 raw image:ORIGINAL signed integer=84;modulus=17;exact ratio=84/17;floor quotient=4

value=q*n+remainder;normalized residue=16 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(14, 7)

Euler ORIGINAL a times unit representative7 = 98

Euler unit representative7 raw image exact quotient := div(98, 17)

Euler unit representative7 raw image exact quotient = 98/17

Euler unit representative7 raw image floor quotient multiple := mul(5, 17)

Euler unit representative7 raw image floor quotient multiple = 85

Euler unit representative7 raw image normalized remainder := sub(98, 85)

Euler unit representative7 raw image normalized remainder = 13

Euler unit representative7 raw image:ORIGINAL signed integer=98;modulus=17;exact ratio=98/17;floor quotient=5

value=q*n+remainder;normalized residue=13 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(14, 8)

Euler ORIGINAL a times unit representative8 = 112

Euler unit representative8 raw image exact quotient := div(112, 17)

Euler unit representative8 raw image exact quotient = 112/17

Euler unit representative8 raw image floor quotient multiple := mul(6, 17)

Euler unit representative8 raw image floor quotient multiple = 102

Euler unit representative8 raw image normalized remainder := sub(112, 102)

Euler unit representative8 raw image normalized remainder = 10

Euler unit representative8 raw image:ORIGINAL signed integer=112;modulus=17;exact ratio=112/17;floor quotient=6

value=q*n+remainder;normalized residue=10 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(14, 9)

Euler ORIGINAL a times unit representative9 = 126

Euler unit representative9 raw image exact quotient := div(126, 17)

Euler unit representative9 raw image exact quotient = 126/17

Euler unit representative9 raw image floor quotient multiple := mul(7, 17)

Euler unit representative9 raw image floor quotient multiple = 119

Euler unit representative9 raw image normalized remainder := sub(126, 119)

Euler unit representative9 raw image normalized remainder = 7

Euler unit representative9 raw image:ORIGINAL signed integer=126;modulus=17;exact ratio=126/17;floor quotient=7

value=q*n+remainder;normalized residue=7 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative10 := mul(14, 10)

Euler ORIGINAL a times unit representative10 = 140

Euler unit representative10 raw image exact quotient := div(140, 17)

Euler unit representative10 raw image exact quotient = 140/17

Euler unit representative10 raw image floor quotient multiple := mul(8, 17)

Euler unit representative10 raw image floor quotient multiple = 136

Euler unit representative10 raw image normalized remainder := sub(140, 136)

Euler unit representative10 raw image normalized remainder = 4

Euler unit representative10 raw image:ORIGINAL signed integer=140;modulus=17;exact ratio=140/17;floor quotient=8

value=q*n+remainder;normalized residue=4 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(14, 11)

Euler ORIGINAL a times unit representative11 = 154

Euler unit representative11 raw image exact quotient := div(154, 17)

Euler unit representative11 raw image exact quotient = 154/17

Euler unit representative11 raw image floor quotient multiple := mul(9, 17)

Euler unit representative11 raw image floor quotient multiple = 153

Euler unit representative11 raw image normalized remainder := sub(154, 153)

Euler unit representative11 raw image normalized remainder = 1

Euler unit representative11 raw image:ORIGINAL signed integer=154;modulus=17;exact ratio=154/17;floor quotient=9

value=q*n+remainder;normalized residue=1 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative12 := mul(14, 12)

Euler ORIGINAL a times unit representative12 = 168

Euler unit representative12 raw image exact quotient := div(168, 17)

Euler unit representative12 raw image exact quotient = 168/17

Euler unit representative12 raw image floor quotient multiple := mul(9, 17)

Euler unit representative12 raw image floor quotient multiple = 153

Euler unit representative12 raw image normalized remainder := sub(168, 153)

Euler unit representative12 raw image normalized remainder = 15

Euler unit representative12 raw image:ORIGINAL signed integer=168;modulus=17;exact ratio=168/17;floor quotient=9

value=q*n+remainder;normalized residue=15 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative13 := mul(14, 13)

Euler ORIGINAL a times unit representative13 = 182

Euler unit representative13 raw image exact quotient := div(182, 17)

Euler unit representative13 raw image exact quotient = 182/17

Euler unit representative13 raw image floor quotient multiple := mul(10, 17)

Euler unit representative13 raw image floor quotient multiple = 170

Euler unit representative13 raw image normalized remainder := sub(182, 170)

Euler unit representative13 raw image normalized remainder = 12

Euler unit representative13 raw image:ORIGINAL signed integer=182;modulus=17;exact ratio=182/17;floor quotient=10

value=q*n+remainder;normalized residue=12 in[0,17);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative14 = 196

Euler unit representative14 raw image exact quotient := div(196, 17)

Euler unit representative14 raw image exact quotient = 196/17

Euler unit representative14 raw image floor quotient multiple := mul(11, 17)

Euler unit representative14 raw image floor quotient multiple = 187

Euler unit representative14 raw image normalized remainder := sub(196, 187)

Euler unit representative14 raw image normalized remainder = 9

Euler unit representative14 raw image:ORIGINAL signed integer=196;modulus=17;exact ratio=196/17;floor quotient=11

value=q*n+remainder;normalized residue=9 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative15 := mul(14, 15)

Euler ORIGINAL a times unit representative15 = 210

Euler unit representative15 raw image exact quotient := div(210, 17)

Euler unit representative15 raw image exact quotient = 210/17

Euler unit representative15 raw image floor quotient multiple := mul(12, 17)

Euler unit representative15 raw image floor quotient multiple = 204

Euler unit representative15 raw image normalized remainder := sub(210, 204)

Euler unit representative15 raw image normalized remainder = 6

Euler unit representative15 raw image:ORIGINAL signed integer=210;modulus=17;exact ratio=210/17;floor quotient=12

value=q*n+remainder;normalized residue=6 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative16 := mul(14, 16)

Euler ORIGINAL a times unit representative16 = 224

Euler unit representative16 raw image exact quotient := div(224, 17)

Euler unit representative16 raw image exact quotient = 224/17

Euler unit representative16 raw image floor quotient multiple := mul(13, 17)

Euler unit representative16 raw image floor quotient multiple = 221

Euler unit representative16 raw image normalized remainder := sub(224, 221)

Euler unit representative16 raw image normalized remainder = 3

Euler unit representative16 raw image:ORIGINAL signed integer=224;modulus=17;exact ratio=224/17;floor quotient=13

value=q*n+remainder;normalized residue=3 in[0,17);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];actual images in ORIGINAL unit order=[14, 11, 8, 5, 2, 16, 13, 10, 7, 4, 1, 15, 12, 9, 6, 3]

SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S112 · 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。
• S111 · 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。
• S110 · premise_and_result_audit — 以精確有理數重算 Euler unit representative16 raw image normalized remainder，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Euler unit representative16 raw image floor quotient multiple，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 Euler unit representative16 raw image exact quotient，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative16，不以小數近似替代。
• S106 · 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。
• S105 · premise_and_result_audit — 以精確有理數重算 Euler unit representative15 raw image normalized remainder，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Euler unit representative15 raw image floor quotient multiple，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Euler unit representative15 raw image exact quotient，不以小數近似替代。
• S102 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative15，不以小數近似替代。
• S101 · 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。
• S100 · 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。
• S099 · 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。
• S098 · 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。
• S097 · 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。
• S096 · 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。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• S086 · 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。
• S085 · 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。
• S084 · 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。
• S083 · 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。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• 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 Euler unit representative5 raw image normalized remainder，不以小數近似替代。
• S054 · 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。
• S053 · 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。
• S052 · 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。
• 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 — 以精確有理數重算 Euler unit representative4 raw image normalized remainder，不以小數近似替代。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• 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 — 以精確有理數重算 Euler unit representative3 raw image normalized remainder，不以小數近似替代。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• 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 — 以精確有理數重算 Euler unit representative2 raw image normalized remainder，不以小數近似替代。
• S039 · 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。
• S038 · 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。
• S037 · 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。
• 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 — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，不以小數近似替代。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• 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 — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，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 — 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 — 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, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 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。