第 20093 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":22,"a":19}。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":22,"a":19}。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(19, 22)

Euler ORIGINAL signed base normalization exact quotient = 19/22

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 22)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(19, 0)

Euler ORIGINAL signed base normalization normalized remainder = 19

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=19;modulus=22;exact ratio=19/22;floor quotient=0

value=q*n+remainder;normalized residue=19 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[19, 22];nonnegative initial pair=[19, 22]

ALL exact Euclidean rows[left,right,q,r]=[[19, 22, 0, 19], [22, 19, 1, 3], [19, 3, 6, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 22];nonnegative initial pair=[0, 22]

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

Euler residue representative1:ORIGINAL gcd inputs=[1, 22];nonnegative initial pair=[1, 22]

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

Euler residue representative2:ORIGINAL gcd inputs=[2, 22];nonnegative initial pair=[2, 22]

ALL exact Euclidean rows[left,right,q,r]=[[2, 22, 0, 2], [22, 2, 11, 0]];terminal gcd=2

Euler residue representative3:ORIGINAL gcd inputs=[3, 22];nonnegative initial pair=[3, 22]

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

Euler residue representative4:ORIGINAL gcd inputs=[4, 22];nonnegative initial pair=[4, 22]

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

Euler residue representative5:ORIGINAL gcd inputs=[5, 22];nonnegative initial pair=[5, 22]

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

Euler residue representative6:ORIGINAL gcd inputs=[6, 22];nonnegative initial pair=[6, 22]

ALL exact Euclidean rows[left,right,q,r]=[[6, 22, 0, 6], [22, 6, 3, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative7:ORIGINAL gcd inputs=[7, 22];nonnegative initial pair=[7, 22]

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

Euler residue representative8:ORIGINAL gcd inputs=[8, 22];nonnegative initial pair=[8, 22]

ALL exact Euclidean rows[left,right,q,r]=[[8, 22, 0, 8], [22, 8, 2, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal gcd=2

Euler residue representative9:ORIGINAL gcd inputs=[9, 22];nonnegative initial pair=[9, 22]

ALL exact Euclidean rows[left,right,q,r]=[[9, 22, 0, 9], [22, 9, 2, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 22];nonnegative initial pair=[10, 22]

ALL exact Euclidean rows[left,right,q,r]=[[10, 22, 0, 10], [22, 10, 2, 2], [10, 2, 5, 0]];terminal gcd=2

Euler residue representative11:ORIGINAL gcd inputs=[11, 22];nonnegative initial pair=[11, 22]

ALL exact Euclidean rows[left,right,q,r]=[[11, 22, 0, 11], [22, 11, 2, 0]];terminal gcd=11

Euler residue representative12:ORIGINAL gcd inputs=[12, 22];nonnegative initial pair=[12, 22]

ALL exact Euclidean rows[left,right,q,r]=[[12, 22, 0, 12], [22, 12, 1, 10], [12, 10, 1, 2], [10, 2, 5, 0]];terminal gcd=2

Euler residue representative13:ORIGINAL gcd inputs=[13, 22];nonnegative initial pair=[13, 22]

ALL exact Euclidean rows[left,right,q,r]=[[13, 22, 0, 13], [22, 13, 1, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 22];nonnegative initial pair=[14, 22]

ALL exact Euclidean rows[left,right,q,r]=[[14, 22, 0, 14], [22, 14, 1, 8], [14, 8, 1, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal gcd=2

Euler residue representative15:ORIGINAL gcd inputs=[15, 22];nonnegative initial pair=[15, 22]

ALL exact Euclidean rows[left,right,q,r]=[[15, 22, 0, 15], [22, 15, 1, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 22];nonnegative initial pair=[16, 22]

ALL exact Euclidean rows[left,right,q,r]=[[16, 22, 0, 16], [22, 16, 1, 6], [16, 6, 2, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative17:ORIGINAL gcd inputs=[17, 22];nonnegative initial pair=[17, 22]

ALL exact Euclidean rows[left,right,q,r]=[[17, 22, 0, 17], [22, 17, 1, 5], [17, 5, 3, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 22];nonnegative initial pair=[18, 22]

ALL exact Euclidean rows[left,right,q,r]=[[18, 22, 0, 18], [22, 18, 1, 4], [18, 4, 4, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative19:ORIGINAL gcd inputs=[19, 22];nonnegative initial pair=[19, 22]

ALL exact Euclidean rows[left,right,q,r]=[[19, 22, 0, 19], [22, 19, 1, 3], [19, 3, 6, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative20:ORIGINAL gcd inputs=[20, 22];nonnegative initial pair=[20, 22]

ALL exact Euclidean rows[left,right,q,r]=[[20, 22, 0, 20], [22, 20, 1, 2], [20, 2, 10, 0]];terminal gcd=2

Euler residue representative21:ORIGINAL gcd inputs=[21, 22];nonnegative initial pair=[21, 22]

ALL exact Euclidean rows[left,right,q,r]=[[21, 22, 0, 21], [22, 21, 1, 1], [21, 1, 21, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 10

COMPLETE original residue carrier0..21;ALL [representative,gcd,unit] rows=[[0, 22, 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, 11, 否], [12, 2, 否], [13, 1, 是], [14, 2, 否], [15, 1, 是], [16, 2, 否], [17, 1, 是], [18, 2, 否], [19, 1, 是], [20, 2, 否], [21, 1, 是]]

increasing COMPLETE U(n)=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21];phi(n)=10;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(19, 7)

Euler ACTUAL base-unit inverse product = 133

Euler ACTUAL inverse check exact quotient := div(133, 22)

Euler ACTUAL inverse check exact quotient = 133/22

Euler ACTUAL inverse check floor quotient multiple := mul(6, 22)

Euler ACTUAL inverse check floor quotient multiple = 132

Euler ACTUAL inverse check normalized remainder := sub(133, 132)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=133;modulus=22;exact ratio=133/22;floor quotient=6

value=q*n+remainder;normalized residue=1 in[0,22);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=22;gcd(original a=19,n)=1;normalized base=19;ACTUAL inverse=7;COMPLETE U(n)=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21]

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(19, 1)

Euler ORIGINAL a times unit representative1 = 19

Euler unit representative1 raw image exact quotient := div(19, 22)

Euler unit representative1 raw image exact quotient = 19/22

Euler unit representative1 raw image floor quotient multiple := mul(0, 22)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(19, 0)

Euler unit representative1 raw image normalized remainder = 19

Euler unit representative1 raw image:ORIGINAL signed integer=19;modulus=22;exact ratio=19/22;floor quotient=0

value=q*n+remainder;normalized residue=19 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(19, 3)

Euler ORIGINAL a times unit representative3 = 57

Euler unit representative3 raw image exact quotient := div(57, 22)

Euler unit representative3 raw image exact quotient = 57/22

Euler unit representative3 raw image floor quotient multiple := mul(2, 22)

Euler unit representative3 raw image floor quotient multiple = 44

Euler unit representative3 raw image normalized remainder := sub(57, 44)

Euler unit representative3 raw image normalized remainder = 13

Euler unit representative3 raw image:ORIGINAL signed integer=57;modulus=22;exact ratio=57/22;floor quotient=2

value=q*n+remainder;normalized residue=13 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(19, 5)

Euler ORIGINAL a times unit representative5 = 95

Euler unit representative5 raw image exact quotient := div(95, 22)

Euler unit representative5 raw image exact quotient = 95/22

Euler unit representative5 raw image floor quotient multiple := mul(4, 22)

Euler unit representative5 raw image floor quotient multiple = 88

Euler unit representative5 raw image normalized remainder := sub(95, 88)

Euler unit representative5 raw image normalized remainder = 7

Euler unit representative5 raw image:ORIGINAL signed integer=95;modulus=22;exact ratio=95/22;floor quotient=4

value=q*n+remainder;normalized residue=7 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(19, 7)

Euler ORIGINAL a times unit representative7 = 133

Euler unit representative7 raw image exact quotient := div(133, 22)

Euler unit representative7 raw image exact quotient = 133/22

Euler unit representative7 raw image floor quotient multiple := mul(6, 22)

Euler unit representative7 raw image floor quotient multiple = 132

Euler unit representative7 raw image normalized remainder := sub(133, 132)

Euler unit representative7 raw image normalized remainder = 1

Euler unit representative7 raw image:ORIGINAL signed integer=133;modulus=22;exact ratio=133/22;floor quotient=6

value=q*n+remainder;normalized residue=1 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(19, 9)

Euler ORIGINAL a times unit representative9 = 171

Euler unit representative9 raw image exact quotient := div(171, 22)

Euler unit representative9 raw image exact quotient = 171/22

Euler unit representative9 raw image floor quotient multiple := mul(7, 22)

Euler unit representative9 raw image floor quotient multiple = 154

Euler unit representative9 raw image normalized remainder := sub(171, 154)

Euler unit representative9 raw image normalized remainder = 17

Euler unit representative9 raw image:ORIGINAL signed integer=171;modulus=22;exact ratio=171/22;floor quotient=7

value=q*n+remainder;normalized residue=17 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative13 := mul(19, 13)

Euler ORIGINAL a times unit representative13 = 247

Euler unit representative13 raw image exact quotient := div(247, 22)

Euler unit representative13 raw image exact quotient = 247/22

Euler unit representative13 raw image floor quotient multiple := mul(11, 22)

Euler unit representative13 raw image floor quotient multiple = 242

Euler unit representative13 raw image normalized remainder := sub(247, 242)

Euler unit representative13 raw image normalized remainder = 5

Euler unit representative13 raw image:ORIGINAL signed integer=247;modulus=22;exact ratio=247/22;floor quotient=11

value=q*n+remainder;normalized residue=5 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative15 := mul(19, 15)

Euler ORIGINAL a times unit representative15 = 285

Euler unit representative15 raw image exact quotient := div(285, 22)

Euler unit representative15 raw image exact quotient = 285/22

Euler unit representative15 raw image floor quotient multiple := mul(12, 22)

Euler unit representative15 raw image floor quotient multiple = 264

Euler unit representative15 raw image normalized remainder := sub(285, 264)

Euler unit representative15 raw image normalized remainder = 21

Euler unit representative15 raw image:ORIGINAL signed integer=285;modulus=22;exact ratio=285/22;floor quotient=12

value=q*n+remainder;normalized residue=21 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative17 := mul(19, 17)

Euler ORIGINAL a times unit representative17 = 323

Euler unit representative17 raw image exact quotient := div(323, 22)

Euler unit representative17 raw image exact quotient = 323/22

Euler unit representative17 raw image floor quotient multiple := mul(14, 22)

Euler unit representative17 raw image floor quotient multiple = 308

Euler unit representative17 raw image normalized remainder := sub(323, 308)

Euler unit representative17 raw image normalized remainder = 15

Euler unit representative17 raw image:ORIGINAL signed integer=323;modulus=22;exact ratio=323/22;floor quotient=14

value=q*n+remainder;normalized residue=15 in[0,22);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative19 = 361

Euler unit representative19 raw image exact quotient := div(361, 22)

Euler unit representative19 raw image exact quotient = 361/22

Euler unit representative19 raw image floor quotient multiple := mul(16, 22)

Euler unit representative19 raw image floor quotient multiple = 352

Euler unit representative19 raw image normalized remainder := sub(361, 352)

Euler unit representative19 raw image normalized remainder = 9

Euler unit representative19 raw image:ORIGINAL signed integer=361;modulus=22;exact ratio=361/22;floor quotient=16

value=q*n+remainder;normalized residue=9 in[0,22);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative21 := mul(19, 21)

Euler ORIGINAL a times unit representative21 = 399

Euler unit representative21 raw image exact quotient := div(399, 22)

Euler unit representative21 raw image exact quotient = 399/22

Euler unit representative21 raw image floor quotient multiple := mul(18, 22)

Euler unit representative21 raw image floor quotient multiple = 396

Euler unit representative21 raw image normalized remainder := sub(399, 396)

Euler unit representative21 raw image normalized remainder = 3

Euler unit representative21 raw image:ORIGINAL signed integer=399;modulus=22;exact ratio=399/22;floor quotient=18

value=q*n+remainder;normalized residue=3 in[0,22);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21];actual images in ORIGINAL unit order=[19, 13, 7, 1, 17, 5, 21, 15, 9, 3]

SOURCE output sorts the mapped list numerically=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S087 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S086 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S085 · premise_and_result_audit — 以精確有理數重算 Euler unit representative21 raw image normalized remainder，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Euler unit representative21 raw image floor quotient multiple，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Euler unit representative21 raw image exact quotient，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative21，不以小數近似替代。
• S081 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S080 · premise_and_result_audit — 以精確有理數重算 Euler unit representative19 raw image normalized remainder，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Euler unit representative19 raw image floor quotient multiple，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Euler unit representative19 raw image exact quotient，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative19，不以小數近似替代。
• S076 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S075 · premise_and_result_audit — 以精確有理數重算 Euler unit representative17 raw image normalized remainder，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Euler unit representative17 raw image floor quotient multiple，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Euler unit representative17 raw image exact quotient，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative17，不以小數近似替代。
• S071 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S070 · premise_and_result_audit — 以精確有理數重算 Euler unit representative15 raw image normalized remainder，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Euler unit representative15 raw image floor quotient multiple，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Euler unit representative15 raw image exact quotient，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative15，不以小數近似替代。
• S066 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S065 · premise_and_result_audit — 以精確有理數重算 Euler unit representative13 raw image normalized remainder，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Euler unit representative13 raw image floor quotient multiple，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Euler unit representative13 raw image exact quotient，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative13，不以小數近似替代。
• S061 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S060 · premise_and_result_audit — 以精確有理數重算 Euler unit representative9 raw image normalized remainder，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Euler unit representative9 raw image floor quotient multiple，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Euler unit representative9 raw image exact quotient，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative9，不以小數近似替代。
• S056 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S055 · premise_and_result_audit — 以精確有理數重算 Euler unit representative7 raw image normalized remainder，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Euler unit representative7 raw image floor quotient multiple，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Euler unit representative7 raw image exact quotient，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative7，不以小數近似替代。
• S051 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S050 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image normalized remainder，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image floor quotient multiple，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image exact quotient，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative5，不以小數近似替代。
• S046 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S045 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image normalized remainder，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image floor quotient multiple，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image exact quotient，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative3，不以小數近似替代。
• S041 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image floor quotient multiple，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image exact quotient，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative1，不以小數近似替代。
• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 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。