第 20005 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":9,"a":7}。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":9,"a":7}。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(7, 9)

Euler ORIGINAL signed base normalization exact quotient = 7/9

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 9)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(7, 0)

Euler ORIGINAL signed base normalization normalized remainder = 7

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=7;modulus=9;exact ratio=7/9;floor quotient=0

value=q*n+remainder;normalized residue=7 in[0,9);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[7, 9];nonnegative initial pair=[7, 9]

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

Euler residue representative0:ORIGINAL gcd inputs=[0, 9];nonnegative initial pair=[0, 9]

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

Euler residue representative1:ORIGINAL gcd inputs=[1, 9];nonnegative initial pair=[1, 9]

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

Euler residue representative2:ORIGINAL gcd inputs=[2, 9];nonnegative initial pair=[2, 9]

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

Euler residue representative3:ORIGINAL gcd inputs=[3, 9];nonnegative initial pair=[3, 9]

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

Euler residue representative4:ORIGINAL gcd inputs=[4, 9];nonnegative initial pair=[4, 9]

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

Euler residue representative5:ORIGINAL gcd inputs=[5, 9];nonnegative initial pair=[5, 9]

ALL exact Euclidean rows[left,right,q,r]=[[5, 9, 0, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 9];nonnegative initial pair=[6, 9]

ALL exact Euclidean rows[left,right,q,r]=[[6, 9, 0, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative7:ORIGINAL gcd inputs=[7, 9];nonnegative initial pair=[7, 9]

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

Euler residue representative8:ORIGINAL gcd inputs=[8, 9];nonnegative initial pair=[8, 9]

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

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

Euler complete reduced residue count phi = 6

COMPLETE original residue carrier0..8;ALL [representative,gcd,unit] rows=[[0, 9, 否], [1, 1, 是], [2, 1, 是], [3, 3, 否], [4, 1, 是], [5, 1, 是], [6, 3, 否], [7, 1, 是], [8, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 4, 5, 7, 8];phi(n)=6;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

ORIGINAL normalized base=7;COMPLETE totient exponent=6;modulus=9

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

Euler power bit0 base square := mul(7, 7)

Euler power bit0 base square = 49

Euler power bit0 squared-base reduction exact quotient := div(49, 9)

Euler power bit0 squared-base reduction exact quotient = 49/9

Euler power bit0 squared-base reduction floor quotient multiple := mul(5, 9)

Euler power bit0 squared-base reduction floor quotient multiple = 45

Euler power bit0 squared-base reduction normalized remainder := sub(49, 45)

Euler power bit0 squared-base reduction normalized remainder = 4

Euler power bit0 squared-base reduction:ORIGINAL signed integer=49;modulus=9;exact ratio=49/9;floor quotient=5

value=q*n+remainder;normalized residue=4 in[0,9);same congruence class,NOT absolute-value replacement

step=0;previous[accumulator,base,exponent]=[1, 7, 6];low bit=0

next[accumulator,base,exponent]=[1, 4, 3];split exponent=2*q+bit preserves original-power invariant

Euler power bit1 accumulator product := mul(1, 4)

Euler power bit1 accumulator product = 4

Euler power bit1 accumulator reduction exact quotient := div(4, 9)

Euler power bit1 accumulator reduction exact quotient = 4/9

Euler power bit1 accumulator reduction floor quotient multiple := mul(0, 9)

Euler power bit1 accumulator reduction floor quotient multiple = 0

Euler power bit1 accumulator reduction normalized remainder := sub(4, 0)

Euler power bit1 accumulator reduction normalized remainder = 4

Euler power bit1 accumulator reduction:ORIGINAL signed integer=4;modulus=9;exact ratio=4/9;floor quotient=0

value=q*n+remainder;normalized residue=4 in[0,9);same congruence class,NOT absolute-value replacement

Euler power bit1 base square := mul(4, 4)

Euler power bit1 base square = 16

Euler power bit1 squared-base reduction exact quotient := div(16, 9)

Euler power bit1 squared-base reduction exact quotient = 16/9

Euler power bit1 squared-base reduction floor quotient multiple := mul(1, 9)

Euler power bit1 squared-base reduction floor quotient multiple = 9

Euler power bit1 squared-base reduction normalized remainder := sub(16, 9)

Euler power bit1 squared-base reduction normalized remainder = 7

Euler power bit1 squared-base reduction:ORIGINAL signed integer=16;modulus=9;exact ratio=16/9;floor quotient=1

value=q*n+remainder;normalized residue=7 in[0,9);same congruence class,NOT absolute-value replacement

step=1;previous[accumulator,base,exponent]=[1, 4, 3];low bit=1

next[accumulator,base,exponent]=[4, 7, 1];split exponent=2*q+bit preserves original-power invariant

Euler power bit2 accumulator product := mul(4, 7)

Euler power bit2 accumulator product = 28

Euler power bit2 accumulator reduction exact quotient := div(28, 9)

Euler power bit2 accumulator reduction exact quotient = 28/9

Euler power bit2 accumulator reduction floor quotient multiple := mul(3, 9)

Euler power bit2 accumulator reduction floor quotient multiple = 27

Euler power bit2 accumulator reduction normalized remainder := sub(28, 27)

Euler power bit2 accumulator reduction normalized remainder = 1

Euler power bit2 accumulator reduction:ORIGINAL signed integer=28;modulus=9;exact ratio=28/9;floor quotient=3

value=q*n+remainder;normalized residue=1 in[0,9);same congruence class,NOT absolute-value replacement

Euler power bit2 base square := mul(7, 7)

Euler power bit2 base square = 49

Euler power bit2 squared-base reduction exact quotient := div(49, 9)

Euler power bit2 squared-base reduction exact quotient = 49/9

Euler power bit2 squared-base reduction floor quotient multiple := mul(5, 9)

Euler power bit2 squared-base reduction floor quotient multiple = 45

Euler power bit2 squared-base reduction normalized remainder := sub(49, 45)

Euler power bit2 squared-base reduction normalized remainder = 4

Euler power bit2 squared-base reduction:ORIGINAL signed integer=49;modulus=9;exact ratio=49/9;floor quotient=5

value=q*n+remainder;normalized residue=4 in[0,9);same congruence class,NOT absolute-value replacement

step=2;previous[accumulator,base,exponent]=[4, 7, 1];low bit=1

next[accumulator,base,exponent]=[1, 4, 0];split exponent=2*q+bit preserves original-power invariant

remaining exponent0;final normalized accumulator=1

EXACT original normalized base^6 modulo9=1

Euler ACTUAL base-unit inverse product := mul(7, 4)

Euler ACTUAL base-unit inverse product = 28

Euler ACTUAL inverse check exact quotient := div(28, 9)

Euler ACTUAL inverse check exact quotient = 28/9

Euler ACTUAL inverse check floor quotient multiple := mul(3, 9)

Euler ACTUAL inverse check floor quotient multiple = 27

Euler ACTUAL inverse check normalized remainder := sub(28, 27)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=28;modulus=9;exact ratio=28/9;floor quotient=3

value=q*n+remainder;normalized residue=1 in[0,9);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=9;gcd(original a=7,n)=1;normalized base=7;ACTUAL inverse=4;COMPLETE U(n)=[1, 2, 4, 5, 7, 8]

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, 4, 5, 7, 8] of size phi=6;multiplication by original a=7 permutes it

multiply ALL unit images:a^(6)*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 modulo9

逐步反向核對

• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• 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 — 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 power bit2 squared-base reduction normalized remainder，不以小數近似替代。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• 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 power bit2 accumulator reduction normalized remainder，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 accumulator reduction 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 power bit2 accumulator reduction exact quotient，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 accumulator product，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 power bit1 squared-base reduction normalized remainder，不以小數近似替代。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• 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 power bit1 accumulator reduction normalized remainder，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator reduction 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 power bit1 accumulator reduction exact quotient，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator 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 — 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 — 以精確有理數重算 Euler power bit0 squared-base reduction normalized remainder，不以小數近似替代。
• S021 · 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。
• S020 · 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。
• S019 · 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。
• 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 — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• 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

(9, 7, 6, 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。