第 20092 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":21,"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":21,"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, 21)

Euler ORIGINAL signed base normalization exact quotient = 19/21

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 21)

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=21;exact ratio=19/21;floor quotient=0

value=q*n+remainder;normalized residue=19 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[19, 21];nonnegative initial pair=[19, 21]

ALL exact Euclidean rows[left,right,q,r]=[[19, 21, 0, 19], [21, 19, 1, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 21];nonnegative initial pair=[0, 21]

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

Euler residue representative1:ORIGINAL gcd inputs=[1, 21];nonnegative initial pair=[1, 21]

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

Euler residue representative2:ORIGINAL gcd inputs=[2, 21];nonnegative initial pair=[2, 21]

ALL exact Euclidean rows[left,right,q,r]=[[2, 21, 0, 2], [21, 2, 10, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 21];nonnegative initial pair=[3, 21]

ALL exact Euclidean rows[left,right,q,r]=[[3, 21, 0, 3], [21, 3, 7, 0]];terminal gcd=3

Euler residue representative4:ORIGINAL gcd inputs=[4, 21];nonnegative initial pair=[4, 21]

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

Euler residue representative5:ORIGINAL gcd inputs=[5, 21];nonnegative initial pair=[5, 21]

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

Euler residue representative6:ORIGINAL gcd inputs=[6, 21];nonnegative initial pair=[6, 21]

ALL exact Euclidean rows[left,right,q,r]=[[6, 21, 0, 6], [21, 6, 3, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative7:ORIGINAL gcd inputs=[7, 21];nonnegative initial pair=[7, 21]

ALL exact Euclidean rows[left,right,q,r]=[[7, 21, 0, 7], [21, 7, 3, 0]];terminal gcd=7

Euler residue representative8:ORIGINAL gcd inputs=[8, 21];nonnegative initial pair=[8, 21]

ALL exact Euclidean rows[left,right,q,r]=[[8, 21, 0, 8], [21, 8, 2, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 21];nonnegative initial pair=[9, 21]

ALL exact Euclidean rows[left,right,q,r]=[[9, 21, 0, 9], [21, 9, 2, 3], [9, 3, 3, 0]];terminal gcd=3

Euler residue representative10:ORIGINAL gcd inputs=[10, 21];nonnegative initial pair=[10, 21]

ALL exact Euclidean rows[left,right,q,r]=[[10, 21, 0, 10], [21, 10, 2, 1], [10, 1, 10, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 21];nonnegative initial pair=[11, 21]

ALL exact Euclidean rows[left,right,q,r]=[[11, 21, 0, 11], [21, 11, 1, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 21];nonnegative initial pair=[12, 21]

ALL exact Euclidean rows[left,right,q,r]=[[12, 21, 0, 12], [21, 12, 1, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal gcd=3

Euler residue representative13:ORIGINAL gcd inputs=[13, 21];nonnegative initial pair=[13, 21]

ALL exact Euclidean rows[left,right,q,r]=[[13, 21, 0, 13], [21, 13, 1, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 21];nonnegative initial pair=[14, 21]

ALL exact Euclidean rows[left,right,q,r]=[[14, 21, 0, 14], [21, 14, 1, 7], [14, 7, 2, 0]];terminal gcd=7

Euler residue representative15:ORIGINAL gcd inputs=[15, 21];nonnegative initial pair=[15, 21]

ALL exact Euclidean rows[left,right,q,r]=[[15, 21, 0, 15], [21, 15, 1, 6], [15, 6, 2, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative16:ORIGINAL gcd inputs=[16, 21];nonnegative initial pair=[16, 21]

ALL exact Euclidean rows[left,right,q,r]=[[16, 21, 0, 16], [21, 16, 1, 5], [16, 5, 3, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative17:ORIGINAL gcd inputs=[17, 21];nonnegative initial pair=[17, 21]

ALL exact Euclidean rows[left,right,q,r]=[[17, 21, 0, 17], [21, 17, 1, 4], [17, 4, 4, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 21];nonnegative initial pair=[18, 21]

ALL exact Euclidean rows[left,right,q,r]=[[18, 21, 0, 18], [21, 18, 1, 3], [18, 3, 6, 0]];terminal gcd=3

Euler residue representative19:ORIGINAL gcd inputs=[19, 21];nonnegative initial pair=[19, 21]

ALL exact Euclidean rows[left,right,q,r]=[[19, 21, 0, 19], [21, 19, 1, 2], [19, 2, 9, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative20:ORIGINAL gcd inputs=[20, 21];nonnegative initial pair=[20, 21]

ALL exact Euclidean rows[left,right,q,r]=[[20, 21, 0, 20], [21, 20, 1, 1], [20, 1, 20, 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..20;ALL [representative,gcd,unit] rows=[[0, 21, 否], [1, 1, 是], [2, 1, 是], [3, 3, 否], [4, 1, 是], [5, 1, 是], [6, 3, 否], [7, 7, 否], [8, 1, 是], [9, 3, 否], [10, 1, 是], [11, 1, 是], [12, 3, 否], [13, 1, 是], [14, 7, 否], [15, 3, 否], [16, 1, 是], [17, 1, 是], [18, 3, 否], [19, 1, 是], [20, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20];phi(n)=12;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(19, 10)

Euler ACTUAL base-unit inverse product = 190

Euler ACTUAL inverse check exact quotient := div(190, 21)

Euler ACTUAL inverse check exact quotient = 190/21

Euler ACTUAL inverse check floor quotient multiple := mul(9, 21)

Euler ACTUAL inverse check floor quotient multiple = 189

Euler ACTUAL inverse check normalized remainder := sub(190, 189)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=190;modulus=21;exact ratio=190/21;floor quotient=9

value=q*n+remainder;normalized residue=1 in[0,21);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=21;gcd(original a=19,n)=1;normalized base=19;ACTUAL inverse=10;COMPLETE U(n)=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]

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, 21)

Euler unit representative1 raw image exact quotient = 19/21

Euler unit representative1 raw image floor quotient multiple := mul(0, 21)

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=21;exact ratio=19/21;floor quotient=0

value=q*n+remainder;normalized residue=19 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(19, 2)

Euler ORIGINAL a times unit representative2 = 38

Euler unit representative2 raw image exact quotient := div(38, 21)

Euler unit representative2 raw image exact quotient = 38/21

Euler unit representative2 raw image floor quotient multiple := mul(1, 21)

Euler unit representative2 raw image floor quotient multiple = 21

Euler unit representative2 raw image normalized remainder := sub(38, 21)

Euler unit representative2 raw image normalized remainder = 17

Euler unit representative2 raw image:ORIGINAL signed integer=38;modulus=21;exact ratio=38/21;floor quotient=1

value=q*n+remainder;normalized residue=17 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(19, 4)

Euler ORIGINAL a times unit representative4 = 76

Euler unit representative4 raw image exact quotient := div(76, 21)

Euler unit representative4 raw image exact quotient = 76/21

Euler unit representative4 raw image floor quotient multiple := mul(3, 21)

Euler unit representative4 raw image floor quotient multiple = 63

Euler unit representative4 raw image normalized remainder := sub(76, 63)

Euler unit representative4 raw image normalized remainder = 13

Euler unit representative4 raw image:ORIGINAL signed integer=76;modulus=21;exact ratio=76/21;floor quotient=3

value=q*n+remainder;normalized residue=13 in[0,21);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, 21)

Euler unit representative5 raw image exact quotient = 95/21

Euler unit representative5 raw image floor quotient multiple := mul(4, 21)

Euler unit representative5 raw image floor quotient multiple = 84

Euler unit representative5 raw image normalized remainder := sub(95, 84)

Euler unit representative5 raw image normalized remainder = 11

Euler unit representative5 raw image:ORIGINAL signed integer=95;modulus=21;exact ratio=95/21;floor quotient=4

value=q*n+remainder;normalized residue=11 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(19, 8)

Euler ORIGINAL a times unit representative8 = 152

Euler unit representative8 raw image exact quotient := div(152, 21)

Euler unit representative8 raw image exact quotient = 152/21

Euler unit representative8 raw image floor quotient multiple := mul(7, 21)

Euler unit representative8 raw image floor quotient multiple = 147

Euler unit representative8 raw image normalized remainder := sub(152, 147)

Euler unit representative8 raw image normalized remainder = 5

Euler unit representative8 raw image:ORIGINAL signed integer=152;modulus=21;exact ratio=152/21;floor quotient=7

value=q*n+remainder;normalized residue=5 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative10 := mul(19, 10)

Euler ORIGINAL a times unit representative10 = 190

Euler unit representative10 raw image exact quotient := div(190, 21)

Euler unit representative10 raw image exact quotient = 190/21

Euler unit representative10 raw image floor quotient multiple := mul(9, 21)

Euler unit representative10 raw image floor quotient multiple = 189

Euler unit representative10 raw image normalized remainder := sub(190, 189)

Euler unit representative10 raw image normalized remainder = 1

Euler unit representative10 raw image:ORIGINAL signed integer=190;modulus=21;exact ratio=190/21;floor quotient=9

value=q*n+remainder;normalized residue=1 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(19, 11)

Euler ORIGINAL a times unit representative11 = 209

Euler unit representative11 raw image exact quotient := div(209, 21)

Euler unit representative11 raw image exact quotient = 209/21

Euler unit representative11 raw image floor quotient multiple := mul(9, 21)

Euler unit representative11 raw image floor quotient multiple = 189

Euler unit representative11 raw image normalized remainder := sub(209, 189)

Euler unit representative11 raw image normalized remainder = 20

Euler unit representative11 raw image:ORIGINAL signed integer=209;modulus=21;exact ratio=209/21;floor quotient=9

value=q*n+remainder;normalized residue=20 in[0,21);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, 21)

Euler unit representative13 raw image exact quotient = 247/21

Euler unit representative13 raw image floor quotient multiple := mul(11, 21)

Euler unit representative13 raw image floor quotient multiple = 231

Euler unit representative13 raw image normalized remainder := sub(247, 231)

Euler unit representative13 raw image normalized remainder = 16

Euler unit representative13 raw image:ORIGINAL signed integer=247;modulus=21;exact ratio=247/21;floor quotient=11

value=q*n+remainder;normalized residue=16 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative16 := mul(19, 16)

Euler ORIGINAL a times unit representative16 = 304

Euler unit representative16 raw image exact quotient := div(304, 21)

Euler unit representative16 raw image exact quotient = 304/21

Euler unit representative16 raw image floor quotient multiple := mul(14, 21)

Euler unit representative16 raw image floor quotient multiple = 294

Euler unit representative16 raw image normalized remainder := sub(304, 294)

Euler unit representative16 raw image normalized remainder = 10

Euler unit representative16 raw image:ORIGINAL signed integer=304;modulus=21;exact ratio=304/21;floor quotient=14

value=q*n+remainder;normalized residue=10 in[0,21);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, 21)

Euler unit representative17 raw image exact quotient = 323/21

Euler unit representative17 raw image floor quotient multiple := mul(15, 21)

Euler unit representative17 raw image floor quotient multiple = 315

Euler unit representative17 raw image normalized remainder := sub(323, 315)

Euler unit representative17 raw image normalized remainder = 8

Euler unit representative17 raw image:ORIGINAL signed integer=323;modulus=21;exact ratio=323/21;floor quotient=15

value=q*n+remainder;normalized residue=8 in[0,21);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, 21)

Euler unit representative19 raw image exact quotient = 361/21

Euler unit representative19 raw image floor quotient multiple := mul(17, 21)

Euler unit representative19 raw image floor quotient multiple = 357

Euler unit representative19 raw image normalized remainder := sub(361, 357)

Euler unit representative19 raw image normalized remainder = 4

Euler unit representative19 raw image:ORIGINAL signed integer=361;modulus=21;exact ratio=361/21;floor quotient=17

value=q*n+remainder;normalized residue=4 in[0,21);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative20 := mul(19, 20)

Euler ORIGINAL a times unit representative20 = 380

Euler unit representative20 raw image exact quotient := div(380, 21)

Euler unit representative20 raw image exact quotient = 380/21

Euler unit representative20 raw image floor quotient multiple := mul(18, 21)

Euler unit representative20 raw image floor quotient multiple = 378

Euler unit representative20 raw image normalized remainder := sub(380, 378)

Euler unit representative20 raw image normalized remainder = 2

Euler unit representative20 raw image:ORIGINAL signed integer=380;modulus=21;exact ratio=380/21;floor quotient=18

value=q*n+remainder;normalized residue=2 in[0,21);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20];actual images in ORIGINAL unit order=[19, 17, 13, 11, 5, 1, 20, 16, 10, 8, 4, 2]

SOURCE output sorts the mapped list numerically=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S096 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S095 · premise_and_result_audit — 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 representative20 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 representative20 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 representative20 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 representative20，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 representative19 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 representative19 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 representative19 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 representative19，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 representative17 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 representative17 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 representative17 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 representative17，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 representative16 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 representative16 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 representative16 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 representative16，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 representative13 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 representative13 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 representative13 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 representative13，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 representative10 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 representative10 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 representative10 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 representative10，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 representative8 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 representative8 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 representative8 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 representative8，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 representative4 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 representative4 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 representative4 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 representative4，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 representative2 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 representative2 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 representative2 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 representative2，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 — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，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 floor quotient multiple，不以小數近似替代。
• S037 · 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。
• S036 · 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。
• 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 — 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 ACTUAL inverse check normalized remainder，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 floor quotient multiple，不以小數近似替代。
• S031 · 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。
• S030 · 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。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，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 — 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, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 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。