第 20096 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":25,"a":22}。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":25,"a":22}。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(22, 25)

Euler ORIGINAL signed base normalization exact quotient = 22/25

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 25)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(22, 0)

Euler ORIGINAL signed base normalization normalized remainder = 22

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=22;modulus=25;exact ratio=22/25;floor quotient=0

value=q*n+remainder;normalized residue=22 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[22, 25];nonnegative initial pair=[22, 25]

ALL exact Euclidean rows[left,right,q,r]=[[22, 25, 0, 22], [25, 22, 1, 3], [22, 3, 7, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 25];nonnegative initial pair=[0, 25]

ALL exact Euclidean rows[left,right,q,r]=[[0, 25, 0, 0]];terminal gcd=25

Euler residue representative1:ORIGINAL gcd inputs=[1, 25];nonnegative initial pair=[1, 25]

ALL exact Euclidean rows[left,right,q,r]=[[1, 25, 0, 1], [25, 1, 25, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 25];nonnegative initial pair=[2, 25]

ALL exact Euclidean rows[left,right,q,r]=[[2, 25, 0, 2], [25, 2, 12, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 25];nonnegative initial pair=[3, 25]

ALL exact Euclidean rows[left,right,q,r]=[[3, 25, 0, 3], [25, 3, 8, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 25];nonnegative initial pair=[4, 25]

ALL exact Euclidean rows[left,right,q,r]=[[4, 25, 0, 4], [25, 4, 6, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 25];nonnegative initial pair=[5, 25]

ALL exact Euclidean rows[left,right,q,r]=[[5, 25, 0, 5], [25, 5, 5, 0]];terminal gcd=5

Euler residue representative6:ORIGINAL gcd inputs=[6, 25];nonnegative initial pair=[6, 25]

ALL exact Euclidean rows[left,right,q,r]=[[6, 25, 0, 6], [25, 6, 4, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 25];nonnegative initial pair=[7, 25]

ALL exact Euclidean rows[left,right,q,r]=[[7, 25, 0, 7], [25, 7, 3, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 25];nonnegative initial pair=[8, 25]

ALL exact Euclidean rows[left,right,q,r]=[[8, 25, 0, 8], [25, 8, 3, 1], [8, 1, 8, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 25];nonnegative initial pair=[9, 25]

ALL exact Euclidean rows[left,right,q,r]=[[9, 25, 0, 9], [25, 9, 2, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 25];nonnegative initial pair=[10, 25]

ALL exact Euclidean rows[left,right,q,r]=[[10, 25, 0, 10], [25, 10, 2, 5], [10, 5, 2, 0]];terminal gcd=5

Euler residue representative11:ORIGINAL gcd inputs=[11, 25];nonnegative initial pair=[11, 25]

ALL exact Euclidean rows[left,right,q,r]=[[11, 25, 0, 11], [25, 11, 2, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 25];nonnegative initial pair=[12, 25]

ALL exact Euclidean rows[left,right,q,r]=[[12, 25, 0, 12], [25, 12, 2, 1], [12, 1, 12, 0]];terminal gcd=1

Euler residue representative13:ORIGINAL gcd inputs=[13, 25];nonnegative initial pair=[13, 25]

ALL exact Euclidean rows[left,right,q,r]=[[13, 25, 0, 13], [25, 13, 1, 12], [13, 12, 1, 1], [12, 1, 12, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 25];nonnegative initial pair=[14, 25]

ALL exact Euclidean rows[left,right,q,r]=[[14, 25, 0, 14], [25, 14, 1, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative15:ORIGINAL gcd inputs=[15, 25];nonnegative initial pair=[15, 25]

ALL exact Euclidean rows[left,right,q,r]=[[15, 25, 0, 15], [25, 15, 1, 10], [15, 10, 1, 5], [10, 5, 2, 0]];terminal gcd=5

Euler residue representative16:ORIGINAL gcd inputs=[16, 25];nonnegative initial pair=[16, 25]

ALL exact Euclidean rows[left,right,q,r]=[[16, 25, 0, 16], [25, 16, 1, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative17:ORIGINAL gcd inputs=[17, 25];nonnegative initial pair=[17, 25]

ALL exact Euclidean rows[left,right,q,r]=[[17, 25, 0, 17], [25, 17, 1, 8], [17, 8, 2, 1], [8, 1, 8, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 25];nonnegative initial pair=[18, 25]

ALL exact Euclidean rows[left,right,q,r]=[[18, 25, 0, 18], [25, 18, 1, 7], [18, 7, 2, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative19:ORIGINAL gcd inputs=[19, 25];nonnegative initial pair=[19, 25]

ALL exact Euclidean rows[left,right,q,r]=[[19, 25, 0, 19], [25, 19, 1, 6], [19, 6, 3, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative20:ORIGINAL gcd inputs=[20, 25];nonnegative initial pair=[20, 25]

ALL exact Euclidean rows[left,right,q,r]=[[20, 25, 0, 20], [25, 20, 1, 5], [20, 5, 4, 0]];terminal gcd=5

Euler residue representative21:ORIGINAL gcd inputs=[21, 25];nonnegative initial pair=[21, 25]

ALL exact Euclidean rows[left,right,q,r]=[[21, 25, 0, 21], [25, 21, 1, 4], [21, 4, 5, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative22:ORIGINAL gcd inputs=[22, 25];nonnegative initial pair=[22, 25]

ALL exact Euclidean rows[left,right,q,r]=[[22, 25, 0, 22], [25, 22, 1, 3], [22, 3, 7, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative23:ORIGINAL gcd inputs=[23, 25];nonnegative initial pair=[23, 25]

ALL exact Euclidean rows[left,right,q,r]=[[23, 25, 0, 23], [25, 23, 1, 2], [23, 2, 11, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative24:ORIGINAL gcd inputs=[24, 25];nonnegative initial pair=[24, 25]

ALL exact Euclidean rows[left,right,q,r]=[[24, 25, 0, 24], [25, 24, 1, 1], [24, 1, 24, 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, 1, 1)

Euler complete reduced residue count phi = 20

COMPLETE original residue carrier0..24;ALL [representative,gcd,unit] rows=[[0, 25, 否], [1, 1, 是], [2, 1, 是], [3, 1, 是], [4, 1, 是], [5, 5, 否], [6, 1, 是], [7, 1, 是], [8, 1, 是], [9, 1, 是], [10, 5, 否], [11, 1, 是], [12, 1, 是], [13, 1, 是], [14, 1, 是], [15, 5, 否], [16, 1, 是], [17, 1, 是], [18, 1, 是], [19, 1, 是], [20, 5, 否], [21, 1, 是], [22, 1, 是], [23, 1, 是], [24, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24];phi(n)=20;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(22, 8)

Euler ACTUAL base-unit inverse product = 176

Euler ACTUAL inverse check exact quotient := div(176, 25)

Euler ACTUAL inverse check exact quotient = 176/25

Euler ACTUAL inverse check floor quotient multiple := mul(7, 25)

Euler ACTUAL inverse check floor quotient multiple = 175

Euler ACTUAL inverse check normalized remainder := sub(176, 175)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=176;modulus=25;exact ratio=176/25;floor quotient=7

value=q*n+remainder;normalized residue=1 in[0,25);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=25;gcd(original a=22,n)=1;normalized base=22;ACTUAL inverse=8;COMPLETE U(n)=[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24]

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(22, 1)

Euler ORIGINAL a times unit representative1 = 22

Euler unit representative1 raw image exact quotient := div(22, 25)

Euler unit representative1 raw image exact quotient = 22/25

Euler unit representative1 raw image floor quotient multiple := mul(0, 25)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(22, 0)

Euler unit representative1 raw image normalized remainder = 22

Euler unit representative1 raw image:ORIGINAL signed integer=22;modulus=25;exact ratio=22/25;floor quotient=0

value=q*n+remainder;normalized residue=22 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(22, 2)

Euler ORIGINAL a times unit representative2 = 44

Euler unit representative2 raw image exact quotient := div(44, 25)

Euler unit representative2 raw image exact quotient = 44/25

Euler unit representative2 raw image floor quotient multiple := mul(1, 25)

Euler unit representative2 raw image floor quotient multiple = 25

Euler unit representative2 raw image normalized remainder := sub(44, 25)

Euler unit representative2 raw image normalized remainder = 19

Euler unit representative2 raw image:ORIGINAL signed integer=44;modulus=25;exact ratio=44/25;floor quotient=1

value=q*n+remainder;normalized residue=19 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(22, 3)

Euler ORIGINAL a times unit representative3 = 66

Euler unit representative3 raw image exact quotient := div(66, 25)

Euler unit representative3 raw image exact quotient = 66/25

Euler unit representative3 raw image floor quotient multiple := mul(2, 25)

Euler unit representative3 raw image floor quotient multiple = 50

Euler unit representative3 raw image normalized remainder := sub(66, 50)

Euler unit representative3 raw image normalized remainder = 16

Euler unit representative3 raw image:ORIGINAL signed integer=66;modulus=25;exact ratio=66/25;floor quotient=2

value=q*n+remainder;normalized residue=16 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(22, 4)

Euler ORIGINAL a times unit representative4 = 88

Euler unit representative4 raw image exact quotient := div(88, 25)

Euler unit representative4 raw image exact quotient = 88/25

Euler unit representative4 raw image floor quotient multiple := mul(3, 25)

Euler unit representative4 raw image floor quotient multiple = 75

Euler unit representative4 raw image normalized remainder := sub(88, 75)

Euler unit representative4 raw image normalized remainder = 13

Euler unit representative4 raw image:ORIGINAL signed integer=88;modulus=25;exact ratio=88/25;floor quotient=3

value=q*n+remainder;normalized residue=13 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative6 := mul(22, 6)

Euler ORIGINAL a times unit representative6 = 132

Euler unit representative6 raw image exact quotient := div(132, 25)

Euler unit representative6 raw image exact quotient = 132/25

Euler unit representative6 raw image floor quotient multiple := mul(5, 25)

Euler unit representative6 raw image floor quotient multiple = 125

Euler unit representative6 raw image normalized remainder := sub(132, 125)

Euler unit representative6 raw image normalized remainder = 7

Euler unit representative6 raw image:ORIGINAL signed integer=132;modulus=25;exact ratio=132/25;floor quotient=5

value=q*n+remainder;normalized residue=7 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(22, 7)

Euler ORIGINAL a times unit representative7 = 154

Euler unit representative7 raw image exact quotient := div(154, 25)

Euler unit representative7 raw image exact quotient = 154/25

Euler unit representative7 raw image floor quotient multiple := mul(6, 25)

Euler unit representative7 raw image floor quotient multiple = 150

Euler unit representative7 raw image normalized remainder := sub(154, 150)

Euler unit representative7 raw image normalized remainder = 4

Euler unit representative7 raw image:ORIGINAL signed integer=154;modulus=25;exact ratio=154/25;floor quotient=6

value=q*n+remainder;normalized residue=4 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(22, 8)

Euler ORIGINAL a times unit representative8 = 176

Euler unit representative8 raw image exact quotient := div(176, 25)

Euler unit representative8 raw image exact quotient = 176/25

Euler unit representative8 raw image floor quotient multiple := mul(7, 25)

Euler unit representative8 raw image floor quotient multiple = 175

Euler unit representative8 raw image normalized remainder := sub(176, 175)

Euler unit representative8 raw image normalized remainder = 1

Euler unit representative8 raw image:ORIGINAL signed integer=176;modulus=25;exact ratio=176/25;floor quotient=7

value=q*n+remainder;normalized residue=1 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(22, 9)

Euler ORIGINAL a times unit representative9 = 198

Euler unit representative9 raw image exact quotient := div(198, 25)

Euler unit representative9 raw image exact quotient = 198/25

Euler unit representative9 raw image floor quotient multiple := mul(7, 25)

Euler unit representative9 raw image floor quotient multiple = 175

Euler unit representative9 raw image normalized remainder := sub(198, 175)

Euler unit representative9 raw image normalized remainder = 23

Euler unit representative9 raw image:ORIGINAL signed integer=198;modulus=25;exact ratio=198/25;floor quotient=7

value=q*n+remainder;normalized residue=23 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(22, 11)

Euler ORIGINAL a times unit representative11 = 242

Euler unit representative11 raw image exact quotient := div(242, 25)

Euler unit representative11 raw image exact quotient = 242/25

Euler unit representative11 raw image floor quotient multiple := mul(9, 25)

Euler unit representative11 raw image floor quotient multiple = 225

Euler unit representative11 raw image normalized remainder := sub(242, 225)

Euler unit representative11 raw image normalized remainder = 17

Euler unit representative11 raw image:ORIGINAL signed integer=242;modulus=25;exact ratio=242/25;floor quotient=9

value=q*n+remainder;normalized residue=17 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative12 := mul(22, 12)

Euler ORIGINAL a times unit representative12 = 264

Euler unit representative12 raw image exact quotient := div(264, 25)

Euler unit representative12 raw image exact quotient = 264/25

Euler unit representative12 raw image floor quotient multiple := mul(10, 25)

Euler unit representative12 raw image floor quotient multiple = 250

Euler unit representative12 raw image normalized remainder := sub(264, 250)

Euler unit representative12 raw image normalized remainder = 14

Euler unit representative12 raw image:ORIGINAL signed integer=264;modulus=25;exact ratio=264/25;floor quotient=10

value=q*n+remainder;normalized residue=14 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative13 := mul(22, 13)

Euler ORIGINAL a times unit representative13 = 286

Euler unit representative13 raw image exact quotient := div(286, 25)

Euler unit representative13 raw image exact quotient = 286/25

Euler unit representative13 raw image floor quotient multiple := mul(11, 25)

Euler unit representative13 raw image floor quotient multiple = 275

Euler unit representative13 raw image normalized remainder := sub(286, 275)

Euler unit representative13 raw image normalized remainder = 11

Euler unit representative13 raw image:ORIGINAL signed integer=286;modulus=25;exact ratio=286/25;floor quotient=11

value=q*n+remainder;normalized residue=11 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative14 := mul(22, 14)

Euler ORIGINAL a times unit representative14 = 308

Euler unit representative14 raw image exact quotient := div(308, 25)

Euler unit representative14 raw image exact quotient = 308/25

Euler unit representative14 raw image floor quotient multiple := mul(12, 25)

Euler unit representative14 raw image floor quotient multiple = 300

Euler unit representative14 raw image normalized remainder := sub(308, 300)

Euler unit representative14 raw image normalized remainder = 8

Euler unit representative14 raw image:ORIGINAL signed integer=308;modulus=25;exact ratio=308/25;floor quotient=12

value=q*n+remainder;normalized residue=8 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative16 := mul(22, 16)

Euler ORIGINAL a times unit representative16 = 352

Euler unit representative16 raw image exact quotient := div(352, 25)

Euler unit representative16 raw image exact quotient = 352/25

Euler unit representative16 raw image floor quotient multiple := mul(14, 25)

Euler unit representative16 raw image floor quotient multiple = 350

Euler unit representative16 raw image normalized remainder := sub(352, 350)

Euler unit representative16 raw image normalized remainder = 2

Euler unit representative16 raw image:ORIGINAL signed integer=352;modulus=25;exact ratio=352/25;floor quotient=14

value=q*n+remainder;normalized residue=2 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative17 := mul(22, 17)

Euler ORIGINAL a times unit representative17 = 374

Euler unit representative17 raw image exact quotient := div(374, 25)

Euler unit representative17 raw image exact quotient = 374/25

Euler unit representative17 raw image floor quotient multiple := mul(14, 25)

Euler unit representative17 raw image floor quotient multiple = 350

Euler unit representative17 raw image normalized remainder := sub(374, 350)

Euler unit representative17 raw image normalized remainder = 24

Euler unit representative17 raw image:ORIGINAL signed integer=374;modulus=25;exact ratio=374/25;floor quotient=14

value=q*n+remainder;normalized residue=24 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative18 := mul(22, 18)

Euler ORIGINAL a times unit representative18 = 396

Euler unit representative18 raw image exact quotient := div(396, 25)

Euler unit representative18 raw image exact quotient = 396/25

Euler unit representative18 raw image floor quotient multiple := mul(15, 25)

Euler unit representative18 raw image floor quotient multiple = 375

Euler unit representative18 raw image normalized remainder := sub(396, 375)

Euler unit representative18 raw image normalized remainder = 21

Euler unit representative18 raw image:ORIGINAL signed integer=396;modulus=25;exact ratio=396/25;floor quotient=15

value=q*n+remainder;normalized residue=21 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative19 := mul(22, 19)

Euler ORIGINAL a times unit representative19 = 418

Euler unit representative19 raw image exact quotient := div(418, 25)

Euler unit representative19 raw image exact quotient = 418/25

Euler unit representative19 raw image floor quotient multiple := mul(16, 25)

Euler unit representative19 raw image floor quotient multiple = 400

Euler unit representative19 raw image normalized remainder := sub(418, 400)

Euler unit representative19 raw image normalized remainder = 18

Euler unit representative19 raw image:ORIGINAL signed integer=418;modulus=25;exact ratio=418/25;floor quotient=16

value=q*n+remainder;normalized residue=18 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative21 := mul(22, 21)

Euler ORIGINAL a times unit representative21 = 462

Euler unit representative21 raw image exact quotient := div(462, 25)

Euler unit representative21 raw image exact quotient = 462/25

Euler unit representative21 raw image floor quotient multiple := mul(18, 25)

Euler unit representative21 raw image floor quotient multiple = 450

Euler unit representative21 raw image normalized remainder := sub(462, 450)

Euler unit representative21 raw image normalized remainder = 12

Euler unit representative21 raw image:ORIGINAL signed integer=462;modulus=25;exact ratio=462/25;floor quotient=18

value=q*n+remainder;normalized residue=12 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative22 := mul(22, 22)

Euler ORIGINAL a times unit representative22 = 484

Euler unit representative22 raw image exact quotient := div(484, 25)

Euler unit representative22 raw image exact quotient = 484/25

Euler unit representative22 raw image floor quotient multiple := mul(19, 25)

Euler unit representative22 raw image floor quotient multiple = 475

Euler unit representative22 raw image normalized remainder := sub(484, 475)

Euler unit representative22 raw image normalized remainder = 9

Euler unit representative22 raw image:ORIGINAL signed integer=484;modulus=25;exact ratio=484/25;floor quotient=19

value=q*n+remainder;normalized residue=9 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative23 := mul(22, 23)

Euler ORIGINAL a times unit representative23 = 506

Euler unit representative23 raw image exact quotient := div(506, 25)

Euler unit representative23 raw image exact quotient = 506/25

Euler unit representative23 raw image floor quotient multiple := mul(20, 25)

Euler unit representative23 raw image floor quotient multiple = 500

Euler unit representative23 raw image normalized remainder := sub(506, 500)

Euler unit representative23 raw image normalized remainder = 6

Euler unit representative23 raw image:ORIGINAL signed integer=506;modulus=25;exact ratio=506/25;floor quotient=20

value=q*n+remainder;normalized residue=6 in[0,25);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative24 := mul(22, 24)

Euler ORIGINAL a times unit representative24 = 528

Euler unit representative24 raw image exact quotient := div(528, 25)

Euler unit representative24 raw image exact quotient = 528/25

Euler unit representative24 raw image floor quotient multiple := mul(21, 25)

Euler unit representative24 raw image floor quotient multiple = 525

Euler unit representative24 raw image normalized remainder := sub(528, 525)

Euler unit representative24 raw image normalized remainder = 3

Euler unit representative24 raw image:ORIGINAL signed integer=528;modulus=25;exact ratio=528/25;floor quotient=21

value=q*n+remainder;normalized residue=3 in[0,25);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24];actual images in ORIGINAL unit order=[22, 19, 16, 13, 7, 4, 1, 23, 17, 14, 11, 8, 2, 24, 21, 18, 12, 9, 6, 3]

SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S140 · 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。
• S139 · 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。
• S138 · premise_and_result_audit — 以精確有理數重算 Euler unit representative24 raw image normalized remainder，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 Euler unit representative24 raw image floor quotient multiple，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 Euler unit representative24 raw image exact quotient，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative24，不以小數近似替代。
• S134 · 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。
• S133 · premise_and_result_audit — 以精確有理數重算 Euler unit representative23 raw image normalized remainder，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 Euler unit representative23 raw image floor quotient multiple，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Euler unit representative23 raw image exact quotient，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative23，不以小數近似替代。
• S129 · 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。
• S128 · premise_and_result_audit — 以精確有理數重算 Euler unit representative22 raw image normalized remainder，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Euler unit representative22 raw image floor quotient multiple，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Euler unit representative22 raw image exact quotient，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative22，不以小數近似替代。
• 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 — 以精確有理數重算 Euler unit representative21 raw image normalized remainder，不以小數近似替代。
• S122 · premise_and_result_audit — 以精確有理數重算 Euler unit representative21 raw image floor quotient multiple，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Euler unit representative21 raw image exact quotient，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative21，不以小數近似替代。
• S119 · 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。
• S118 · premise_and_result_audit — 以精確有理數重算 Euler unit representative19 raw image normalized remainder，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Euler unit representative19 raw image floor quotient multiple，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Euler unit representative19 raw image exact quotient，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative19，不以小數近似替代。
• S114 · 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。
• S113 · 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。
• S112 · 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。
• S111 · 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。
• S110 · 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。
• S109 · 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。
• S108 · 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。
• S107 · 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。
• S106 · 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。
• S105 · 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。
• S104 · 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。
• S103 · 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。
• S102 · 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。
• S101 · 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。
• S100 · 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。
• S099 · 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。
• S098 · 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。
• S097 · 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。
• S096 · 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。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• S086 · 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。
• S085 · 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。
• S084 · 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。
• S083 · 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。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• 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 representative7 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 representative7 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 representative7 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 representative7，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 representative6 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 representative6 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 representative6 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 representative6，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 representative4 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 representative4 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 representative4 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 representative4，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 representative3 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 representative3 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 representative3 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 representative3，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 representative2 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 representative2 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 representative2 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 representative2，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 representative1 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 representative1 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 representative1 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 representative1，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 — 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 ACTUAL inverse check normalized remainder，不以小數近似替代。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S032 · premise_and_result_audit — 以精確有理數重算 Euler complete reduced residue count phi，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 — 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 — 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 — 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, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 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。