第 20013 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-congruence；資料={"n":17,"a":14}。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-congruence；資料={"n":17,"a":14}。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-congruence;original snapshot and mathematics retained

explicit ordered source output fields=[n, normalized_base, totient, residue, congruent_to_one]

Euler ORIGINAL signed base normalization exact quotient := div(14, 17)

Euler ORIGINAL signed base normalization exact quotient = 14/17

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 17)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(14, 0)

Euler ORIGINAL signed base normalization normalized remainder = 14

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=14;modulus=17;exact ratio=14/17;floor quotient=0

value=q*n+remainder;normalized residue=14 in[0,17);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[14, 17];nonnegative initial pair=[14, 17]

ALL exact Euclidean rows[left,right,q,r]=[[14, 17, 0, 14], [17, 14, 1, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 17];nonnegative initial pair=[0, 17]

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

Euler residue representative1:ORIGINAL gcd inputs=[1, 17];nonnegative initial pair=[1, 17]

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

Euler residue representative2:ORIGINAL gcd inputs=[2, 17];nonnegative initial pair=[2, 17]

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

Euler residue representative3:ORIGINAL gcd inputs=[3, 17];nonnegative initial pair=[3, 17]

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

Euler residue representative4:ORIGINAL gcd inputs=[4, 17];nonnegative initial pair=[4, 17]

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

Euler residue representative5:ORIGINAL gcd inputs=[5, 17];nonnegative initial pair=[5, 17]

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

Euler residue representative6:ORIGINAL gcd inputs=[6, 17];nonnegative initial pair=[6, 17]

ALL exact Euclidean rows[left,right,q,r]=[[6, 17, 0, 6], [17, 6, 2, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 17];nonnegative initial pair=[7, 17]

ALL exact Euclidean rows[left,right,q,r]=[[7, 17, 0, 7], [17, 7, 2, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 17];nonnegative initial pair=[8, 17]

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

Euler residue representative9:ORIGINAL gcd inputs=[9, 17];nonnegative initial pair=[9, 17]

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

Euler residue representative10:ORIGINAL gcd inputs=[10, 17];nonnegative initial pair=[10, 17]

ALL exact Euclidean rows[left,right,q,r]=[[10, 17, 0, 10], [17, 10, 1, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 17];nonnegative initial pair=[11, 17]

ALL exact Euclidean rows[left,right,q,r]=[[11, 17, 0, 11], [17, 11, 1, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 17];nonnegative initial pair=[12, 17]

ALL exact Euclidean rows[left,right,q,r]=[[12, 17, 0, 12], [17, 12, 1, 5], [12, 5, 2, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative13:ORIGINAL gcd inputs=[13, 17];nonnegative initial pair=[13, 17]

ALL exact Euclidean rows[left,right,q,r]=[[13, 17, 0, 13], [17, 13, 1, 4], [13, 4, 3, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 17];nonnegative initial pair=[14, 17]

ALL exact Euclidean rows[left,right,q,r]=[[14, 17, 0, 14], [17, 14, 1, 3], [14, 3, 4, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative15:ORIGINAL gcd inputs=[15, 17];nonnegative initial pair=[15, 17]

ALL exact Euclidean rows[left,right,q,r]=[[15, 17, 0, 15], [17, 15, 1, 2], [15, 2, 7, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 17];nonnegative initial pair=[16, 17]

ALL exact Euclidean rows[left,right,q,r]=[[16, 17, 0, 16], [17, 16, 1, 1], [16, 1, 16, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 16

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

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];phi(n)=16;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

ORIGINAL normalized base=14;COMPLETE totient exponent=16;modulus=17

initialize accumulator1,current_base=base,remaining=phi;invariant accumulator*current_base^remaining congruent to the original totient power

Euler power bit0 base square := mul(14, 14)

Euler power bit0 base square = 196

Euler power bit0 squared-base reduction exact quotient := div(196, 17)

Euler power bit0 squared-base reduction exact quotient = 196/17

Euler power bit0 squared-base reduction floor quotient multiple := mul(11, 17)

Euler power bit0 squared-base reduction floor quotient multiple = 187

Euler power bit0 squared-base reduction normalized remainder := sub(196, 187)

Euler power bit0 squared-base reduction normalized remainder = 9

Euler power bit0 squared-base reduction:ORIGINAL signed integer=196;modulus=17;exact ratio=196/17;floor quotient=11

value=q*n+remainder;normalized residue=9 in[0,17);same congruence class,NOT absolute-value replacement

step=0;previous[accumulator,base,exponent]=[1, 14, 16];low bit=0

next[accumulator,base,exponent]=[1, 9, 8];split exponent=2*q+bit preserves original-power invariant

Euler power bit1 base square := mul(9, 9)

Euler power bit1 base square = 81

Euler power bit1 squared-base reduction exact quotient := div(81, 17)

Euler power bit1 squared-base reduction exact quotient = 81/17

Euler power bit1 squared-base reduction floor quotient multiple := mul(4, 17)

Euler power bit1 squared-base reduction floor quotient multiple = 68

Euler power bit1 squared-base reduction normalized remainder := sub(81, 68)

Euler power bit1 squared-base reduction normalized remainder = 13

Euler power bit1 squared-base reduction:ORIGINAL signed integer=81;modulus=17;exact ratio=81/17;floor quotient=4

value=q*n+remainder;normalized residue=13 in[0,17);same congruence class,NOT absolute-value replacement

step=1;previous[accumulator,base,exponent]=[1, 9, 8];low bit=0

next[accumulator,base,exponent]=[1, 13, 4];split exponent=2*q+bit preserves original-power invariant

Euler power bit2 base square := mul(13, 13)

Euler power bit2 base square = 169

Euler power bit2 squared-base reduction exact quotient := div(169, 17)

Euler power bit2 squared-base reduction exact quotient = 169/17

Euler power bit2 squared-base reduction floor quotient multiple := mul(9, 17)

Euler power bit2 squared-base reduction floor quotient multiple = 153

Euler power bit2 squared-base reduction normalized remainder := sub(169, 153)

Euler power bit2 squared-base reduction normalized remainder = 16

Euler power bit2 squared-base reduction:ORIGINAL signed integer=169;modulus=17;exact ratio=169/17;floor quotient=9

value=q*n+remainder;normalized residue=16 in[0,17);same congruence class,NOT absolute-value replacement

step=2;previous[accumulator,base,exponent]=[1, 13, 4];low bit=0

next[accumulator,base,exponent]=[1, 16, 2];split exponent=2*q+bit preserves original-power invariant

Euler power bit3 base square := mul(16, 16)

Euler power bit3 base square = 256

Euler power bit3 squared-base reduction exact quotient := div(256, 17)

Euler power bit3 squared-base reduction exact quotient = 256/17

Euler power bit3 squared-base reduction floor quotient multiple := mul(15, 17)

Euler power bit3 squared-base reduction floor quotient multiple = 255

Euler power bit3 squared-base reduction normalized remainder := sub(256, 255)

Euler power bit3 squared-base reduction normalized remainder = 1

Euler power bit3 squared-base reduction:ORIGINAL signed integer=256;modulus=17;exact ratio=256/17;floor quotient=15

value=q*n+remainder;normalized residue=1 in[0,17);same congruence class,NOT absolute-value replacement

step=3;previous[accumulator,base,exponent]=[1, 16, 2];low bit=0

next[accumulator,base,exponent]=[1, 1, 1];split exponent=2*q+bit preserves original-power invariant

Euler power bit4 accumulator product := mul(1, 1)

Euler power bit4 accumulator product = 1

Euler power bit4 accumulator reduction exact quotient := div(1, 17)

Euler power bit4 accumulator reduction exact quotient = 1/17

Euler power bit4 accumulator reduction floor quotient multiple := mul(0, 17)

Euler power bit4 accumulator reduction floor quotient multiple = 0

Euler power bit4 accumulator reduction normalized remainder := sub(1, 0)

Euler power bit4 accumulator reduction normalized remainder = 1

Euler power bit4 accumulator reduction:ORIGINAL signed integer=1;modulus=17;exact ratio=1/17;floor quotient=0

value=q*n+remainder;normalized residue=1 in[0,17);same congruence class,NOT absolute-value replacement

Euler power bit4 base square := mul(1, 1)

Euler power bit4 base square = 1

Euler power bit4 squared-base reduction exact quotient := div(1, 17)

Euler power bit4 squared-base reduction exact quotient = 1/17

Euler power bit4 squared-base reduction floor quotient multiple := mul(0, 17)

Euler power bit4 squared-base reduction floor quotient multiple = 0

Euler power bit4 squared-base reduction normalized remainder := sub(1, 0)

Euler power bit4 squared-base reduction normalized remainder = 1

Euler power bit4 squared-base reduction:ORIGINAL signed integer=1;modulus=17;exact ratio=1/17;floor quotient=0

value=q*n+remainder;normalized residue=1 in[0,17);same congruence class,NOT absolute-value replacement

step=4;previous[accumulator,base,exponent]=[1, 1, 1];low bit=1

next[accumulator,base,exponent]=[1, 1, 0];split exponent=2*q+bit preserves original-power invariant

remaining exponent0;final normalized accumulator=1

EXACT original normalized base^16 modulo17=1

Euler ACTUAL base-unit inverse product := mul(14, 11)

Euler ACTUAL base-unit inverse product = 154

Euler ACTUAL inverse check exact quotient := div(154, 17)

Euler ACTUAL inverse check exact quotient = 154/17

Euler ACTUAL inverse check floor quotient multiple := mul(9, 17)

Euler ACTUAL inverse check floor quotient multiple = 153

Euler ACTUAL inverse check normalized remainder := sub(154, 153)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=154;modulus=17;exact ratio=154/17;floor quotient=9

value=q*n+remainder;normalized residue=1 in[0,17);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=17;gcd(original a=14,n)=1;normalized base=14;ACTUAL inverse=11;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

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)

SAME complete U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] of size phi=16;multiplication by original a=14 permutes it

multiply ALL unit images:a^(16)*product(U) congruent product(U);EVERY U factor is a UNIT,so product(U) has reverse-product inverse;cancel ONLY this unit to get a^phi congruent1 modulo17

逐步反向核對

• S069 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S068 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• 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 — 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 power bit4 squared-base reduction normalized remainder，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 squared-base reduction 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 power bit4 squared-base reduction exact quotient，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 base square，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 power bit4 accumulator reduction normalized remainder，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 accumulator reduction 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 power bit4 accumulator reduction exact quotient，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 accumulator product，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S048 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction normalized remainder，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction floor quotient multiple，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction exact quotient，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 base square，不以小數近似替代。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction normalized remainder，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction floor quotient multiple，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction exact quotient，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 base square，不以小數近似替代。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction normalized remainder，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction floor quotient multiple，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction exact quotient，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 base square，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 — 以精確有理數重算 Euler power bit0 squared-base reduction normalized remainder，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction floor quotient multiple，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction exact quotient，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 base square，不以小數近似替代。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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

(17, 14, 16, 1, 是)

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。