第 20097 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":26,"a":23}。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":26,"a":23}。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(23, 26)

Euler ORIGINAL signed base normalization exact quotient = 23/26

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 26)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(23, 0)

Euler ORIGINAL signed base normalization normalized remainder = 23

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=23;modulus=26;exact ratio=23/26;floor quotient=0

value=q*n+remainder;normalized residue=23 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[23, 26];nonnegative initial pair=[23, 26]

ALL exact Euclidean rows[left,right,q,r]=[[23, 26, 0, 23], [26, 23, 1, 3], [23, 3, 7, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 26];nonnegative initial pair=[0, 26]

ALL exact Euclidean rows[left,right,q,r]=[[0, 26, 0, 0]];terminal gcd=26

Euler residue representative1:ORIGINAL gcd inputs=[1, 26];nonnegative initial pair=[1, 26]

ALL exact Euclidean rows[left,right,q,r]=[[1, 26, 0, 1], [26, 1, 26, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 26];nonnegative initial pair=[2, 26]

ALL exact Euclidean rows[left,right,q,r]=[[2, 26, 0, 2], [26, 2, 13, 0]];terminal gcd=2

Euler residue representative3:ORIGINAL gcd inputs=[3, 26];nonnegative initial pair=[3, 26]

ALL exact Euclidean rows[left,right,q,r]=[[3, 26, 0, 3], [26, 3, 8, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 26];nonnegative initial pair=[4, 26]

ALL exact Euclidean rows[left,right,q,r]=[[4, 26, 0, 4], [26, 4, 6, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative5:ORIGINAL gcd inputs=[5, 26];nonnegative initial pair=[5, 26]

ALL exact Euclidean rows[left,right,q,r]=[[5, 26, 0, 5], [26, 5, 5, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 26];nonnegative initial pair=[6, 26]

ALL exact Euclidean rows[left,right,q,r]=[[6, 26, 0, 6], [26, 6, 4, 2], [6, 2, 3, 0]];terminal gcd=2

Euler residue representative7:ORIGINAL gcd inputs=[7, 26];nonnegative initial pair=[7, 26]

ALL exact Euclidean rows[left,right,q,r]=[[7, 26, 0, 7], [26, 7, 3, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 26];nonnegative initial pair=[8, 26]

ALL exact Euclidean rows[left,right,q,r]=[[8, 26, 0, 8], [26, 8, 3, 2], [8, 2, 4, 0]];terminal gcd=2

Euler residue representative9:ORIGINAL gcd inputs=[9, 26];nonnegative initial pair=[9, 26]

ALL exact Euclidean rows[left,right,q,r]=[[9, 26, 0, 9], [26, 9, 2, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 26];nonnegative initial pair=[10, 26]

ALL exact Euclidean rows[left,right,q,r]=[[10, 26, 0, 10], [26, 10, 2, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative11:ORIGINAL gcd inputs=[11, 26];nonnegative initial pair=[11, 26]

ALL exact Euclidean rows[left,right,q,r]=[[11, 26, 0, 11], [26, 11, 2, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 26];nonnegative initial pair=[12, 26]

ALL exact Euclidean rows[left,right,q,r]=[[12, 26, 0, 12], [26, 12, 2, 2], [12, 2, 6, 0]];terminal gcd=2

Euler residue representative13:ORIGINAL gcd inputs=[13, 26];nonnegative initial pair=[13, 26]

ALL exact Euclidean rows[left,right,q,r]=[[13, 26, 0, 13], [26, 13, 2, 0]];terminal gcd=13

Euler residue representative14:ORIGINAL gcd inputs=[14, 26];nonnegative initial pair=[14, 26]

ALL exact Euclidean rows[left,right,q,r]=[[14, 26, 0, 14], [26, 14, 1, 12], [14, 12, 1, 2], [12, 2, 6, 0]];terminal gcd=2

Euler residue representative15:ORIGINAL gcd inputs=[15, 26];nonnegative initial pair=[15, 26]

ALL exact Euclidean rows[left,right,q,r]=[[15, 26, 0, 15], [26, 15, 1, 11], [15, 11, 1, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 26];nonnegative initial pair=[16, 26]

ALL exact Euclidean rows[left,right,q,r]=[[16, 26, 0, 16], [26, 16, 1, 10], [16, 10, 1, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative17:ORIGINAL gcd inputs=[17, 26];nonnegative initial pair=[17, 26]

ALL exact Euclidean rows[left,right,q,r]=[[17, 26, 0, 17], [26, 17, 1, 9], [17, 9, 1, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 26];nonnegative initial pair=[18, 26]

ALL exact Euclidean rows[left,right,q,r]=[[18, 26, 0, 18], [26, 18, 1, 8], [18, 8, 2, 2], [8, 2, 4, 0]];terminal gcd=2

Euler residue representative19:ORIGINAL gcd inputs=[19, 26];nonnegative initial pair=[19, 26]

ALL exact Euclidean rows[left,right,q,r]=[[19, 26, 0, 19], [26, 19, 1, 7], [19, 7, 2, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative20:ORIGINAL gcd inputs=[20, 26];nonnegative initial pair=[20, 26]

ALL exact Euclidean rows[left,right,q,r]=[[20, 26, 0, 20], [26, 20, 1, 6], [20, 6, 3, 2], [6, 2, 3, 0]];terminal gcd=2

Euler residue representative21:ORIGINAL gcd inputs=[21, 26];nonnegative initial pair=[21, 26]

ALL exact Euclidean rows[left,right,q,r]=[[21, 26, 0, 21], [26, 21, 1, 5], [21, 5, 4, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative22:ORIGINAL gcd inputs=[22, 26];nonnegative initial pair=[22, 26]

ALL exact Euclidean rows[left,right,q,r]=[[22, 26, 0, 22], [26, 22, 1, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative23:ORIGINAL gcd inputs=[23, 26];nonnegative initial pair=[23, 26]

ALL exact Euclidean rows[left,right,q,r]=[[23, 26, 0, 23], [26, 23, 1, 3], [23, 3, 7, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative24:ORIGINAL gcd inputs=[24, 26];nonnegative initial pair=[24, 26]

ALL exact Euclidean rows[left,right,q,r]=[[24, 26, 0, 24], [26, 24, 1, 2], [24, 2, 12, 0]];terminal gcd=2

Euler residue representative25:ORIGINAL gcd inputs=[25, 26];nonnegative initial pair=[25, 26]

ALL exact Euclidean rows[left,right,q,r]=[[25, 26, 0, 25], [26, 25, 1, 1], [25, 1, 25, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 12

COMPLETE original residue carrier0..25;ALL [representative,gcd,unit] rows=[[0, 26, 否], [1, 1, 是], [2, 2, 否], [3, 1, 是], [4, 2, 否], [5, 1, 是], [6, 2, 否], [7, 1, 是], [8, 2, 否], [9, 1, 是], [10, 2, 否], [11, 1, 是], [12, 2, 否], [13, 13, 否], [14, 2, 否], [15, 1, 是], [16, 2, 否], [17, 1, 是], [18, 2, 否], [19, 1, 是], [20, 2, 否], [21, 1, 是], [22, 2, 否], [23, 1, 是], [24, 2, 否], [25, 1, 是]]

increasing COMPLETE U(n)=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25];phi(n)=12;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(23, 17)

Euler ACTUAL base-unit inverse product = 391

Euler ACTUAL inverse check exact quotient := div(391, 26)

Euler ACTUAL inverse check exact quotient = 391/26

Euler ACTUAL inverse check floor quotient multiple := mul(15, 26)

Euler ACTUAL inverse check floor quotient multiple = 390

Euler ACTUAL inverse check normalized remainder := sub(391, 390)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=391;modulus=26;exact ratio=391/26;floor quotient=15

value=q*n+remainder;normalized residue=1 in[0,26);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=26;gcd(original a=23,n)=1;normalized base=23;ACTUAL inverse=17;COMPLETE U(n)=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25]

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(23, 1)

Euler ORIGINAL a times unit representative1 = 23

Euler unit representative1 raw image exact quotient := div(23, 26)

Euler unit representative1 raw image exact quotient = 23/26

Euler unit representative1 raw image floor quotient multiple := mul(0, 26)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(23, 0)

Euler unit representative1 raw image normalized remainder = 23

Euler unit representative1 raw image:ORIGINAL signed integer=23;modulus=26;exact ratio=23/26;floor quotient=0

value=q*n+remainder;normalized residue=23 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(23, 3)

Euler ORIGINAL a times unit representative3 = 69

Euler unit representative3 raw image exact quotient := div(69, 26)

Euler unit representative3 raw image exact quotient = 69/26

Euler unit representative3 raw image floor quotient multiple := mul(2, 26)

Euler unit representative3 raw image floor quotient multiple = 52

Euler unit representative3 raw image normalized remainder := sub(69, 52)

Euler unit representative3 raw image normalized remainder = 17

Euler unit representative3 raw image:ORIGINAL signed integer=69;modulus=26;exact ratio=69/26;floor quotient=2

value=q*n+remainder;normalized residue=17 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(23, 5)

Euler ORIGINAL a times unit representative5 = 115

Euler unit representative5 raw image exact quotient := div(115, 26)

Euler unit representative5 raw image exact quotient = 115/26

Euler unit representative5 raw image floor quotient multiple := mul(4, 26)

Euler unit representative5 raw image floor quotient multiple = 104

Euler unit representative5 raw image normalized remainder := sub(115, 104)

Euler unit representative5 raw image normalized remainder = 11

Euler unit representative5 raw image:ORIGINAL signed integer=115;modulus=26;exact ratio=115/26;floor quotient=4

value=q*n+remainder;normalized residue=11 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(23, 7)

Euler ORIGINAL a times unit representative7 = 161

Euler unit representative7 raw image exact quotient := div(161, 26)

Euler unit representative7 raw image exact quotient = 161/26

Euler unit representative7 raw image floor quotient multiple := mul(6, 26)

Euler unit representative7 raw image floor quotient multiple = 156

Euler unit representative7 raw image normalized remainder := sub(161, 156)

Euler unit representative7 raw image normalized remainder = 5

Euler unit representative7 raw image:ORIGINAL signed integer=161;modulus=26;exact ratio=161/26;floor quotient=6

value=q*n+remainder;normalized residue=5 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(23, 9)

Euler ORIGINAL a times unit representative9 = 207

Euler unit representative9 raw image exact quotient := div(207, 26)

Euler unit representative9 raw image exact quotient = 207/26

Euler unit representative9 raw image floor quotient multiple := mul(7, 26)

Euler unit representative9 raw image floor quotient multiple = 182

Euler unit representative9 raw image normalized remainder := sub(207, 182)

Euler unit representative9 raw image normalized remainder = 25

Euler unit representative9 raw image:ORIGINAL signed integer=207;modulus=26;exact ratio=207/26;floor quotient=7

value=q*n+remainder;normalized residue=25 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(23, 11)

Euler ORIGINAL a times unit representative11 = 253

Euler unit representative11 raw image exact quotient := div(253, 26)

Euler unit representative11 raw image exact quotient = 253/26

Euler unit representative11 raw image floor quotient multiple := mul(9, 26)

Euler unit representative11 raw image floor quotient multiple = 234

Euler unit representative11 raw image normalized remainder := sub(253, 234)

Euler unit representative11 raw image normalized remainder = 19

Euler unit representative11 raw image:ORIGINAL signed integer=253;modulus=26;exact ratio=253/26;floor quotient=9

value=q*n+remainder;normalized residue=19 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative15 := mul(23, 15)

Euler ORIGINAL a times unit representative15 = 345

Euler unit representative15 raw image exact quotient := div(345, 26)

Euler unit representative15 raw image exact quotient = 345/26

Euler unit representative15 raw image floor quotient multiple := mul(13, 26)

Euler unit representative15 raw image floor quotient multiple = 338

Euler unit representative15 raw image normalized remainder := sub(345, 338)

Euler unit representative15 raw image normalized remainder = 7

Euler unit representative15 raw image:ORIGINAL signed integer=345;modulus=26;exact ratio=345/26;floor quotient=13

value=q*n+remainder;normalized residue=7 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative17 := mul(23, 17)

Euler ORIGINAL a times unit representative17 = 391

Euler unit representative17 raw image exact quotient := div(391, 26)

Euler unit representative17 raw image exact quotient = 391/26

Euler unit representative17 raw image floor quotient multiple := mul(15, 26)

Euler unit representative17 raw image floor quotient multiple = 390

Euler unit representative17 raw image normalized remainder := sub(391, 390)

Euler unit representative17 raw image normalized remainder = 1

Euler unit representative17 raw image:ORIGINAL signed integer=391;modulus=26;exact ratio=391/26;floor quotient=15

value=q*n+remainder;normalized residue=1 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative19 := mul(23, 19)

Euler ORIGINAL a times unit representative19 = 437

Euler unit representative19 raw image exact quotient := div(437, 26)

Euler unit representative19 raw image exact quotient = 437/26

Euler unit representative19 raw image floor quotient multiple := mul(16, 26)

Euler unit representative19 raw image floor quotient multiple = 416

Euler unit representative19 raw image normalized remainder := sub(437, 416)

Euler unit representative19 raw image normalized remainder = 21

Euler unit representative19 raw image:ORIGINAL signed integer=437;modulus=26;exact ratio=437/26;floor quotient=16

value=q*n+remainder;normalized residue=21 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative21 := mul(23, 21)

Euler ORIGINAL a times unit representative21 = 483

Euler unit representative21 raw image exact quotient := div(483, 26)

Euler unit representative21 raw image exact quotient = 483/26

Euler unit representative21 raw image floor quotient multiple := mul(18, 26)

Euler unit representative21 raw image floor quotient multiple = 468

Euler unit representative21 raw image normalized remainder := sub(483, 468)

Euler unit representative21 raw image normalized remainder = 15

Euler unit representative21 raw image:ORIGINAL signed integer=483;modulus=26;exact ratio=483/26;floor quotient=18

value=q*n+remainder;normalized residue=15 in[0,26);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative23 = 529

Euler unit representative23 raw image exact quotient := div(529, 26)

Euler unit representative23 raw image exact quotient = 529/26

Euler unit representative23 raw image floor quotient multiple := mul(20, 26)

Euler unit representative23 raw image floor quotient multiple = 520

Euler unit representative23 raw image normalized remainder := sub(529, 520)

Euler unit representative23 raw image normalized remainder = 9

Euler unit representative23 raw image:ORIGINAL signed integer=529;modulus=26;exact ratio=529/26;floor quotient=20

value=q*n+remainder;normalized residue=9 in[0,26);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative25 := mul(23, 25)

Euler ORIGINAL a times unit representative25 = 575

Euler unit representative25 raw image exact quotient := div(575, 26)

Euler unit representative25 raw image exact quotient = 575/26

Euler unit representative25 raw image floor quotient multiple := mul(22, 26)

Euler unit representative25 raw image floor quotient multiple = 572

Euler unit representative25 raw image normalized remainder := sub(575, 572)

Euler unit representative25 raw image normalized remainder = 3

Euler unit representative25 raw image:ORIGINAL signed integer=575;modulus=26;exact ratio=575/26;floor quotient=22

value=q*n+remainder;normalized residue=3 in[0,26);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25];actual images in ORIGINAL unit order=[23, 17, 11, 5, 25, 19, 7, 1, 21, 15, 9, 3]

SOURCE output sorts the mapped list numerically=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25];full-list equality=是;not a claim each point is fixed

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S099 · premise_and_result_audit — 以精確有理數重算 Euler unit representative25 raw image normalized remainder，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Euler unit representative25 raw image floor quotient multiple，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Euler unit representative25 raw image exact quotient，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative25，不以小數近似替代。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• S086 · 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。
• S085 · 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。
• S084 · 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。
• S083 · 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。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S069 · premise_and_result_audit — 以精確有理數重算 Euler unit representative11 raw image normalized remainder，不以小數近似替代。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S038 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · premise_and_result_audit — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 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, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], 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。