第 20090 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":19,"a":16}。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":19,"a":16}。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(16, 19)

Euler ORIGINAL signed base normalization exact quotient = 16/19

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 19)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(16, 0)

Euler ORIGINAL signed base normalization normalized remainder = 16

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=16;modulus=19;exact ratio=16/19;floor quotient=0

value=q*n+remainder;normalized residue=16 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[16, 19];nonnegative initial pair=[16, 19]

ALL exact Euclidean rows[left,right,q,r]=[[16, 19, 0, 16], [19, 16, 1, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 19];nonnegative initial pair=[0, 19]

ALL exact Euclidean rows[left,right,q,r]=[[0, 19, 0, 0]];terminal gcd=19

Euler residue representative1:ORIGINAL gcd inputs=[1, 19];nonnegative initial pair=[1, 19]

ALL exact Euclidean rows[left,right,q,r]=[[1, 19, 0, 1], [19, 1, 19, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 19];nonnegative initial pair=[2, 19]

ALL exact Euclidean rows[left,right,q,r]=[[2, 19, 0, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 19];nonnegative initial pair=[3, 19]

ALL exact Euclidean rows[left,right,q,r]=[[3, 19, 0, 3], [19, 3, 6, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 19];nonnegative initial pair=[4, 19]

ALL exact Euclidean rows[left,right,q,r]=[[4, 19, 0, 4], [19, 4, 4, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 19];nonnegative initial pair=[5, 19]

ALL exact Euclidean rows[left,right,q,r]=[[5, 19, 0, 5], [19, 5, 3, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 19];nonnegative initial pair=[6, 19]

ALL exact Euclidean rows[left,right,q,r]=[[6, 19, 0, 6], [19, 6, 3, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 19];nonnegative initial pair=[7, 19]

ALL exact Euclidean rows[left,right,q,r]=[[7, 19, 0, 7], [19, 7, 2, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 19];nonnegative initial pair=[8, 19]

ALL exact Euclidean rows[left,right,q,r]=[[8, 19, 0, 8], [19, 8, 2, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 19];nonnegative initial pair=[9, 19]

ALL exact Euclidean rows[left,right,q,r]=[[9, 19, 0, 9], [19, 9, 2, 1], [9, 1, 9, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 19];nonnegative initial pair=[10, 19]

ALL exact Euclidean rows[left,right,q,r]=[[10, 19, 0, 10], [19, 10, 1, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 19];nonnegative initial pair=[11, 19]

ALL exact Euclidean rows[left,right,q,r]=[[11, 19, 0, 11], [19, 11, 1, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 19];nonnegative initial pair=[12, 19]

ALL exact Euclidean rows[left,right,q,r]=[[12, 19, 0, 12], [19, 12, 1, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative13:ORIGINAL gcd inputs=[13, 19];nonnegative initial pair=[13, 19]

ALL exact Euclidean rows[left,right,q,r]=[[13, 19, 0, 13], [19, 13, 1, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 19];nonnegative initial pair=[14, 19]

ALL exact Euclidean rows[left,right,q,r]=[[14, 19, 0, 14], [19, 14, 1, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative15:ORIGINAL gcd inputs=[15, 19];nonnegative initial pair=[15, 19]

ALL exact Euclidean rows[left,right,q,r]=[[15, 19, 0, 15], [19, 15, 1, 4], [15, 4, 3, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 19];nonnegative initial pair=[16, 19]

ALL exact Euclidean rows[left,right,q,r]=[[16, 19, 0, 16], [19, 16, 1, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative17:ORIGINAL gcd inputs=[17, 19];nonnegative initial pair=[17, 19]

ALL exact Euclidean rows[left,right,q,r]=[[17, 19, 0, 17], [19, 17, 1, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 19];nonnegative initial pair=[18, 19]

ALL exact Euclidean rows[left,right,q,r]=[[18, 19, 0, 18], [19, 18, 1, 1], [18, 1, 18, 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, 1, 1)

Euler complete reduced residue count phi = 18

COMPLETE original residue carrier0..18;ALL [representative,gcd,unit] rows=[[0, 19, 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], [17, 1, 是], [18, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];phi(n)=18;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(16, 6)

Euler ACTUAL base-unit inverse product = 96

Euler ACTUAL inverse check exact quotient := div(96, 19)

Euler ACTUAL inverse check exact quotient = 96/19

Euler ACTUAL inverse check floor quotient multiple := mul(5, 19)

Euler ACTUAL inverse check floor quotient multiple = 95

Euler ACTUAL inverse check normalized remainder := sub(96, 95)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=96;modulus=19;exact ratio=96/19;floor quotient=5

value=q*n+remainder;normalized residue=1 in[0,19);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=19;gcd(original a=16,n)=1;normalized base=16;ACTUAL inverse=6;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

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(16, 1)

Euler ORIGINAL a times unit representative1 = 16

Euler unit representative1 raw image exact quotient := div(16, 19)

Euler unit representative1 raw image exact quotient = 16/19

Euler unit representative1 raw image floor quotient multiple := mul(0, 19)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(16, 0)

Euler unit representative1 raw image normalized remainder = 16

Euler unit representative1 raw image:ORIGINAL signed integer=16;modulus=19;exact ratio=16/19;floor quotient=0

value=q*n+remainder;normalized residue=16 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(16, 2)

Euler ORIGINAL a times unit representative2 = 32

Euler unit representative2 raw image exact quotient := div(32, 19)

Euler unit representative2 raw image exact quotient = 32/19

Euler unit representative2 raw image floor quotient multiple := mul(1, 19)

Euler unit representative2 raw image floor quotient multiple = 19

Euler unit representative2 raw image normalized remainder := sub(32, 19)

Euler unit representative2 raw image normalized remainder = 13

Euler unit representative2 raw image:ORIGINAL signed integer=32;modulus=19;exact ratio=32/19;floor quotient=1

value=q*n+remainder;normalized residue=13 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(16, 3)

Euler ORIGINAL a times unit representative3 = 48

Euler unit representative3 raw image exact quotient := div(48, 19)

Euler unit representative3 raw image exact quotient = 48/19

Euler unit representative3 raw image floor quotient multiple := mul(2, 19)

Euler unit representative3 raw image floor quotient multiple = 38

Euler unit representative3 raw image normalized remainder := sub(48, 38)

Euler unit representative3 raw image normalized remainder = 10

Euler unit representative3 raw image:ORIGINAL signed integer=48;modulus=19;exact ratio=48/19;floor quotient=2

value=q*n+remainder;normalized residue=10 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(16, 4)

Euler ORIGINAL a times unit representative4 = 64

Euler unit representative4 raw image exact quotient := div(64, 19)

Euler unit representative4 raw image exact quotient = 64/19

Euler unit representative4 raw image floor quotient multiple := mul(3, 19)

Euler unit representative4 raw image floor quotient multiple = 57

Euler unit representative4 raw image normalized remainder := sub(64, 57)

Euler unit representative4 raw image normalized remainder = 7

Euler unit representative4 raw image:ORIGINAL signed integer=64;modulus=19;exact ratio=64/19;floor quotient=3

value=q*n+remainder;normalized residue=7 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(16, 5)

Euler ORIGINAL a times unit representative5 = 80

Euler unit representative5 raw image exact quotient := div(80, 19)

Euler unit representative5 raw image exact quotient = 80/19

Euler unit representative5 raw image floor quotient multiple := mul(4, 19)

Euler unit representative5 raw image floor quotient multiple = 76

Euler unit representative5 raw image normalized remainder := sub(80, 76)

Euler unit representative5 raw image normalized remainder = 4

Euler unit representative5 raw image:ORIGINAL signed integer=80;modulus=19;exact ratio=80/19;floor quotient=4

value=q*n+remainder;normalized residue=4 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative6 := mul(16, 6)

Euler ORIGINAL a times unit representative6 = 96

Euler unit representative6 raw image exact quotient := div(96, 19)

Euler unit representative6 raw image exact quotient = 96/19

Euler unit representative6 raw image floor quotient multiple := mul(5, 19)

Euler unit representative6 raw image floor quotient multiple = 95

Euler unit representative6 raw image normalized remainder := sub(96, 95)

Euler unit representative6 raw image normalized remainder = 1

Euler unit representative6 raw image:ORIGINAL signed integer=96;modulus=19;exact ratio=96/19;floor quotient=5

value=q*n+remainder;normalized residue=1 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(16, 7)

Euler ORIGINAL a times unit representative7 = 112

Euler unit representative7 raw image exact quotient := div(112, 19)

Euler unit representative7 raw image exact quotient = 112/19

Euler unit representative7 raw image floor quotient multiple := mul(5, 19)

Euler unit representative7 raw image floor quotient multiple = 95

Euler unit representative7 raw image normalized remainder := sub(112, 95)

Euler unit representative7 raw image normalized remainder = 17

Euler unit representative7 raw image:ORIGINAL signed integer=112;modulus=19;exact ratio=112/19;floor quotient=5

value=q*n+remainder;normalized residue=17 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(16, 8)

Euler ORIGINAL a times unit representative8 = 128

Euler unit representative8 raw image exact quotient := div(128, 19)

Euler unit representative8 raw image exact quotient = 128/19

Euler unit representative8 raw image floor quotient multiple := mul(6, 19)

Euler unit representative8 raw image floor quotient multiple = 114

Euler unit representative8 raw image normalized remainder := sub(128, 114)

Euler unit representative8 raw image normalized remainder = 14

Euler unit representative8 raw image:ORIGINAL signed integer=128;modulus=19;exact ratio=128/19;floor quotient=6

value=q*n+remainder;normalized residue=14 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(16, 9)

Euler ORIGINAL a times unit representative9 = 144

Euler unit representative9 raw image exact quotient := div(144, 19)

Euler unit representative9 raw image exact quotient = 144/19

Euler unit representative9 raw image floor quotient multiple := mul(7, 19)

Euler unit representative9 raw image floor quotient multiple = 133

Euler unit representative9 raw image normalized remainder := sub(144, 133)

Euler unit representative9 raw image normalized remainder = 11

Euler unit representative9 raw image:ORIGINAL signed integer=144;modulus=19;exact ratio=144/19;floor quotient=7

value=q*n+remainder;normalized residue=11 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative10 := mul(16, 10)

Euler ORIGINAL a times unit representative10 = 160

Euler unit representative10 raw image exact quotient := div(160, 19)

Euler unit representative10 raw image exact quotient = 160/19

Euler unit representative10 raw image floor quotient multiple := mul(8, 19)

Euler unit representative10 raw image floor quotient multiple = 152

Euler unit representative10 raw image normalized remainder := sub(160, 152)

Euler unit representative10 raw image normalized remainder = 8

Euler unit representative10 raw image:ORIGINAL signed integer=160;modulus=19;exact ratio=160/19;floor quotient=8

value=q*n+remainder;normalized residue=8 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(16, 11)

Euler ORIGINAL a times unit representative11 = 176

Euler unit representative11 raw image exact quotient := div(176, 19)

Euler unit representative11 raw image exact quotient = 176/19

Euler unit representative11 raw image floor quotient multiple := mul(9, 19)

Euler unit representative11 raw image floor quotient multiple = 171

Euler unit representative11 raw image normalized remainder := sub(176, 171)

Euler unit representative11 raw image normalized remainder = 5

Euler unit representative11 raw image:ORIGINAL signed integer=176;modulus=19;exact ratio=176/19;floor quotient=9

value=q*n+remainder;normalized residue=5 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative12 := mul(16, 12)

Euler ORIGINAL a times unit representative12 = 192

Euler unit representative12 raw image exact quotient := div(192, 19)

Euler unit representative12 raw image exact quotient = 192/19

Euler unit representative12 raw image floor quotient multiple := mul(10, 19)

Euler unit representative12 raw image floor quotient multiple = 190

Euler unit representative12 raw image normalized remainder := sub(192, 190)

Euler unit representative12 raw image normalized remainder = 2

Euler unit representative12 raw image:ORIGINAL signed integer=192;modulus=19;exact ratio=192/19;floor quotient=10

value=q*n+remainder;normalized residue=2 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative13 := mul(16, 13)

Euler ORIGINAL a times unit representative13 = 208

Euler unit representative13 raw image exact quotient := div(208, 19)

Euler unit representative13 raw image exact quotient = 208/19

Euler unit representative13 raw image floor quotient multiple := mul(10, 19)

Euler unit representative13 raw image floor quotient multiple = 190

Euler unit representative13 raw image normalized remainder := sub(208, 190)

Euler unit representative13 raw image normalized remainder = 18

Euler unit representative13 raw image:ORIGINAL signed integer=208;modulus=19;exact ratio=208/19;floor quotient=10

value=q*n+remainder;normalized residue=18 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative14 := mul(16, 14)

Euler ORIGINAL a times unit representative14 = 224

Euler unit representative14 raw image exact quotient := div(224, 19)

Euler unit representative14 raw image exact quotient = 224/19

Euler unit representative14 raw image floor quotient multiple := mul(11, 19)

Euler unit representative14 raw image floor quotient multiple = 209

Euler unit representative14 raw image normalized remainder := sub(224, 209)

Euler unit representative14 raw image normalized remainder = 15

Euler unit representative14 raw image:ORIGINAL signed integer=224;modulus=19;exact ratio=224/19;floor quotient=11

value=q*n+remainder;normalized residue=15 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative15 := mul(16, 15)

Euler ORIGINAL a times unit representative15 = 240

Euler unit representative15 raw image exact quotient := div(240, 19)

Euler unit representative15 raw image exact quotient = 240/19

Euler unit representative15 raw image floor quotient multiple := mul(12, 19)

Euler unit representative15 raw image floor quotient multiple = 228

Euler unit representative15 raw image normalized remainder := sub(240, 228)

Euler unit representative15 raw image normalized remainder = 12

Euler unit representative15 raw image:ORIGINAL signed integer=240;modulus=19;exact ratio=240/19;floor quotient=12

value=q*n+remainder;normalized residue=12 in[0,19);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative16 = 256

Euler unit representative16 raw image exact quotient := div(256, 19)

Euler unit representative16 raw image exact quotient = 256/19

Euler unit representative16 raw image floor quotient multiple := mul(13, 19)

Euler unit representative16 raw image floor quotient multiple = 247

Euler unit representative16 raw image normalized remainder := sub(256, 247)

Euler unit representative16 raw image normalized remainder = 9

Euler unit representative16 raw image:ORIGINAL signed integer=256;modulus=19;exact ratio=256/19;floor quotient=13

value=q*n+remainder;normalized residue=9 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative17 := mul(16, 17)

Euler ORIGINAL a times unit representative17 = 272

Euler unit representative17 raw image exact quotient := div(272, 19)

Euler unit representative17 raw image exact quotient = 272/19

Euler unit representative17 raw image floor quotient multiple := mul(14, 19)

Euler unit representative17 raw image floor quotient multiple = 266

Euler unit representative17 raw image normalized remainder := sub(272, 266)

Euler unit representative17 raw image normalized remainder = 6

Euler unit representative17 raw image:ORIGINAL signed integer=272;modulus=19;exact ratio=272/19;floor quotient=14

value=q*n+remainder;normalized residue=6 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative18 := mul(16, 18)

Euler ORIGINAL a times unit representative18 = 288

Euler unit representative18 raw image exact quotient := div(288, 19)

Euler unit representative18 raw image exact quotient = 288/19

Euler unit representative18 raw image floor quotient multiple := mul(15, 19)

Euler unit representative18 raw image floor quotient multiple = 285

Euler unit representative18 raw image normalized remainder := sub(288, 285)

Euler unit representative18 raw image normalized remainder = 3

Euler unit representative18 raw image:ORIGINAL signed integer=288;modulus=19;exact ratio=288/19;floor quotient=15

value=q*n+remainder;normalized residue=3 in[0,19);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, 17, 18];actual images in ORIGINAL unit order=[16, 13, 10, 7, 4, 1, 17, 14, 11, 8, 5, 2, 18, 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, 17, 18];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S124 · 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。
• S123 · 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。
• S122 · premise_and_result_audit — 以精確有理數重算 Euler unit representative18 raw image normalized remainder，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Euler unit representative18 raw image floor quotient multiple，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Euler unit representative18 raw image exact quotient，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative18，不以小數近似替代。
• S118 · 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。
• S117 · premise_and_result_audit — 以精確有理數重算 Euler unit representative17 raw image normalized remainder，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Euler unit representative17 raw image floor quotient multiple，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Euler unit representative17 raw image exact quotient，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative17，不以小數近似替代。
• S113 · 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。
• S112 · 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。
• S111 · 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。
• S110 · 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。
• S109 · 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。
• S108 · 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。
• S107 · 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。
• S106 · 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。
• S105 · 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。
• S104 · 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。
• S103 · 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。
• S102 · 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。
• S101 · 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。
• S100 · 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。
• S099 · 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。
• S098 · 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。
• S097 · 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。
• S096 · 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。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• 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 — 以精確有理數重算 Euler unit representative11 raw image normalized remainder，不以小數近似替代。
• S086 · 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。
• S085 · 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。
• S084 · 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。
• S083 · 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。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• 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 representative5 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 representative5 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 representative5 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 representative5，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 representative4 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 representative4 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 representative4 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 representative4，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 representative3 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 representative3 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 representative3 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 representative3，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 representative2 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 representative2 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 representative2 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 representative2，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 representative1 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 representative1 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 representative1 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 representative1，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 — 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 ACTUAL inverse check normalized remainder，不以小數近似替代。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• 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 complete reduced residue count phi，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 — 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 — 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, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 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。