第 20087 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":16,"a":13}。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":16,"a":13}。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(13, 16)

Euler ORIGINAL signed base normalization exact quotient = 13/16

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 16)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(13, 0)

Euler ORIGINAL signed base normalization normalized remainder = 13

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=13;modulus=16;exact ratio=13/16;floor quotient=0

value=q*n+remainder;normalized residue=13 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[13, 16];nonnegative initial pair=[13, 16]

ALL exact Euclidean rows[left,right,q,r]=[[13, 16, 0, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 16];nonnegative initial pair=[0, 16]

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

Euler residue representative1:ORIGINAL gcd inputs=[1, 16];nonnegative initial pair=[1, 16]

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

Euler residue representative2:ORIGINAL gcd inputs=[2, 16];nonnegative initial pair=[2, 16]

ALL exact Euclidean rows[left,right,q,r]=[[2, 16, 0, 2], [16, 2, 8, 0]];terminal gcd=2

Euler residue representative3:ORIGINAL gcd inputs=[3, 16];nonnegative initial pair=[3, 16]

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

Euler residue representative4:ORIGINAL gcd inputs=[4, 16];nonnegative initial pair=[4, 16]

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

Euler residue representative5:ORIGINAL gcd inputs=[5, 16];nonnegative initial pair=[5, 16]

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

Euler residue representative6:ORIGINAL gcd inputs=[6, 16];nonnegative initial pair=[6, 16]

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

Euler residue representative7:ORIGINAL gcd inputs=[7, 16];nonnegative initial pair=[7, 16]

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

Euler residue representative8:ORIGINAL gcd inputs=[8, 16];nonnegative initial pair=[8, 16]

ALL exact Euclidean rows[left,right,q,r]=[[8, 16, 0, 8], [16, 8, 2, 0]];terminal gcd=8

Euler residue representative9:ORIGINAL gcd inputs=[9, 16];nonnegative initial pair=[9, 16]

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

Euler residue representative10:ORIGINAL gcd inputs=[10, 16];nonnegative initial pair=[10, 16]

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

Euler residue representative11:ORIGINAL gcd inputs=[11, 16];nonnegative initial pair=[11, 16]

ALL exact Euclidean rows[left,right,q,r]=[[11, 16, 0, 11], [16, 11, 1, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 16];nonnegative initial pair=[12, 16]

ALL exact Euclidean rows[left,right,q,r]=[[12, 16, 0, 12], [16, 12, 1, 4], [12, 4, 3, 0]];terminal gcd=4

Euler residue representative13:ORIGINAL gcd inputs=[13, 16];nonnegative initial pair=[13, 16]

ALL exact Euclidean rows[left,right,q,r]=[[13, 16, 0, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 16];nonnegative initial pair=[14, 16]

ALL exact Euclidean rows[left,right,q,r]=[[14, 16, 0, 14], [16, 14, 1, 2], [14, 2, 7, 0]];terminal gcd=2

Euler residue representative15:ORIGINAL gcd inputs=[15, 16];nonnegative initial pair=[15, 16]

ALL exact Euclidean rows[left,right,q,r]=[[15, 16, 0, 15], [16, 15, 1, 1], [15, 1, 15, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 8

COMPLETE original residue carrier0..15;ALL [representative,gcd,unit] rows=[[0, 16, 否], [1, 1, 是], [2, 2, 否], [3, 1, 是], [4, 4, 否], [5, 1, 是], [6, 2, 否], [7, 1, 是], [8, 8, 否], [9, 1, 是], [10, 2, 否], [11, 1, 是], [12, 4, 否], [13, 1, 是], [14, 2, 否], [15, 1, 是]]

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

Euler ACTUAL base-unit inverse product := mul(13, 5)

Euler ACTUAL base-unit inverse product = 65

Euler ACTUAL inverse check exact quotient := div(65, 16)

Euler ACTUAL inverse check exact quotient = 65/16

Euler ACTUAL inverse check floor quotient multiple := mul(4, 16)

Euler ACTUAL inverse check floor quotient multiple = 64

Euler ACTUAL inverse check normalized remainder := sub(65, 64)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=65;modulus=16;exact ratio=65/16;floor quotient=4

value=q*n+remainder;normalized residue=1 in[0,16);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=16;gcd(original a=13,n)=1;normalized base=13;ACTUAL inverse=5;COMPLETE U(n)=[1, 3, 5, 7, 9, 11, 13, 15]

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(13, 1)

Euler ORIGINAL a times unit representative1 = 13

Euler unit representative1 raw image exact quotient := div(13, 16)

Euler unit representative1 raw image exact quotient = 13/16

Euler unit representative1 raw image floor quotient multiple := mul(0, 16)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(13, 0)

Euler unit representative1 raw image normalized remainder = 13

Euler unit representative1 raw image:ORIGINAL signed integer=13;modulus=16;exact ratio=13/16;floor quotient=0

value=q*n+remainder;normalized residue=13 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(13, 3)

Euler ORIGINAL a times unit representative3 = 39

Euler unit representative3 raw image exact quotient := div(39, 16)

Euler unit representative3 raw image exact quotient = 39/16

Euler unit representative3 raw image floor quotient multiple := mul(2, 16)

Euler unit representative3 raw image floor quotient multiple = 32

Euler unit representative3 raw image normalized remainder := sub(39, 32)

Euler unit representative3 raw image normalized remainder = 7

Euler unit representative3 raw image:ORIGINAL signed integer=39;modulus=16;exact ratio=39/16;floor quotient=2

value=q*n+remainder;normalized residue=7 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(13, 5)

Euler ORIGINAL a times unit representative5 = 65

Euler unit representative5 raw image exact quotient := div(65, 16)

Euler unit representative5 raw image exact quotient = 65/16

Euler unit representative5 raw image floor quotient multiple := mul(4, 16)

Euler unit representative5 raw image floor quotient multiple = 64

Euler unit representative5 raw image normalized remainder := sub(65, 64)

Euler unit representative5 raw image normalized remainder = 1

Euler unit representative5 raw image:ORIGINAL signed integer=65;modulus=16;exact ratio=65/16;floor quotient=4

value=q*n+remainder;normalized residue=1 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(13, 7)

Euler ORIGINAL a times unit representative7 = 91

Euler unit representative7 raw image exact quotient := div(91, 16)

Euler unit representative7 raw image exact quotient = 91/16

Euler unit representative7 raw image floor quotient multiple := mul(5, 16)

Euler unit representative7 raw image floor quotient multiple = 80

Euler unit representative7 raw image normalized remainder := sub(91, 80)

Euler unit representative7 raw image normalized remainder = 11

Euler unit representative7 raw image:ORIGINAL signed integer=91;modulus=16;exact ratio=91/16;floor quotient=5

value=q*n+remainder;normalized residue=11 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(13, 9)

Euler ORIGINAL a times unit representative9 = 117

Euler unit representative9 raw image exact quotient := div(117, 16)

Euler unit representative9 raw image exact quotient = 117/16

Euler unit representative9 raw image floor quotient multiple := mul(7, 16)

Euler unit representative9 raw image floor quotient multiple = 112

Euler unit representative9 raw image normalized remainder := sub(117, 112)

Euler unit representative9 raw image normalized remainder = 5

Euler unit representative9 raw image:ORIGINAL signed integer=117;modulus=16;exact ratio=117/16;floor quotient=7

value=q*n+remainder;normalized residue=5 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(13, 11)

Euler ORIGINAL a times unit representative11 = 143

Euler unit representative11 raw image exact quotient := div(143, 16)

Euler unit representative11 raw image exact quotient = 143/16

Euler unit representative11 raw image floor quotient multiple := mul(8, 16)

Euler unit representative11 raw image floor quotient multiple = 128

Euler unit representative11 raw image normalized remainder := sub(143, 128)

Euler unit representative11 raw image normalized remainder = 15

Euler unit representative11 raw image:ORIGINAL signed integer=143;modulus=16;exact ratio=143/16;floor quotient=8

value=q*n+remainder;normalized residue=15 in[0,16);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative13 = 169

Euler unit representative13 raw image exact quotient := div(169, 16)

Euler unit representative13 raw image exact quotient = 169/16

Euler unit representative13 raw image floor quotient multiple := mul(10, 16)

Euler unit representative13 raw image floor quotient multiple = 160

Euler unit representative13 raw image normalized remainder := sub(169, 160)

Euler unit representative13 raw image normalized remainder = 9

Euler unit representative13 raw image:ORIGINAL signed integer=169;modulus=16;exact ratio=169/16;floor quotient=10

value=q*n+remainder;normalized residue=9 in[0,16);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative15 := mul(13, 15)

Euler ORIGINAL a times unit representative15 = 195

Euler unit representative15 raw image exact quotient := div(195, 16)

Euler unit representative15 raw image exact quotient = 195/16

Euler unit representative15 raw image floor quotient multiple := mul(12, 16)

Euler unit representative15 raw image floor quotient multiple = 192

Euler unit representative15 raw image normalized remainder := sub(195, 192)

Euler unit representative15 raw image normalized remainder = 3

Euler unit representative15 raw image:ORIGINAL signed integer=195;modulus=16;exact ratio=195/16;floor quotient=12

value=q*n+remainder;normalized residue=3 in[0,16);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 3, 5, 7, 9, 11, 13, 15];actual images in ORIGINAL unit order=[13, 7, 1, 11, 5, 15, 9, 3]

SOURCE output sorts the mapped list numerically=[1, 3, 5, 7, 9, 11, 13, 15];full-list equality=是;not a claim each point is fixed

逐步反向核對

• 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 — 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 representative15 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 representative15 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 representative15 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 representative15，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 representative13 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 representative13 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 representative13 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 representative13，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 representative11 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 representative11 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 representative11 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 representative11，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 representative9 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 representative9 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 representative9 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 representative9，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 representative7 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 representative7 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 representative7 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 representative7，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 representative5 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 representative5 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 representative5 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 representative5，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 representative3 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 representative3 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 representative3 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 representative3，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 — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，不以小數近似替代。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• 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 — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], 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。