第 19859 題：Fermat小定理

原題與已知條件

exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={"modulus":12,"a":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={"modulus":12,"a":10}。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-fermat-little-theorem

逐步推導

source operation=flt-hypothesis-audit;original snapshot and mathematics retained

explicit ordered source output fields=[modulus, original_base, prime, coprime, coprime_form_applicable, power_modulus_minus_1, boundary]

ACTUAL integer modulus=12;complete trial range2..floor_sqrt=3

checked rows[divisor,quotient,remainder]=[[2, 6, 0]];first exact factor=2;prime=否

original signed base=10;positive modulus=12;use nonnegative gcd inputs=[10, 12]

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

Fermat original signed base exact quotient := div(10, 12)

Fermat original signed base exact quotient = 5/6

Fermat original signed base floor quotient multiple := mul(0, 12)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(10, 0)

Fermat original signed base normalized remainder = 10

Fermat original signed base:signed integer value=10;positive modulus=12;exact quotient=5/6;floor q=0

value=q*modulus+remainder;normalized remainder=10 in[0,12)

Fermat original modular exponent:original normalized base=10;original exponent=11;modulus=12

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

Fermat original modular exponent bit0 accumulator product := mul(1, 10)

Fermat original modular exponent bit0 accumulator product = 10

Fermat original modular exponent bit0 accumulator reduction exact quotient := div(10, 12)

Fermat original modular exponent bit0 accumulator reduction exact quotient = 5/6

Fermat original modular exponent bit0 accumulator reduction floor quotient multiple := mul(0, 12)

Fermat original modular exponent bit0 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit0 accumulator reduction normalized remainder := sub(10, 0)

Fermat original modular exponent bit0 accumulator reduction normalized remainder = 10

Fermat original modular exponent bit0 accumulator reduction:signed integer value=10;positive modulus=12;exact quotient=5/6;floor q=0

value=q*modulus+remainder;normalized remainder=10 in[0,12)

Fermat original modular exponent bit0 current base square := mul(10, 10)

Fermat original modular exponent bit0 current base square = 100

Fermat original modular exponent bit0 squared base reduction exact quotient := div(100, 12)

Fermat original modular exponent bit0 squared base reduction exact quotient = 25/3

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(8, 12)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 96

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(100, 96)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 4

Fermat original modular exponent bit0 squared base reduction:signed integer value=100;positive modulus=12;exact quotient=25/3;floor q=8

value=q*modulus+remainder;normalized remainder=4 in[0,12)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 10, 11];low bit=1

next[accumulator,base,exponent]=[10, 4, 5];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(10, 4)

Fermat original modular exponent bit1 accumulator product = 40

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(40, 12)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 10/3

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(3, 12)

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple = 36

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(40, 36)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 4

Fermat original modular exponent bit1 accumulator reduction:signed integer value=40;positive modulus=12;exact quotient=10/3;floor q=3

value=q*modulus+remainder;normalized remainder=4 in[0,12)

Fermat original modular exponent bit1 current base square := mul(4, 4)

Fermat original modular exponent bit1 current base square = 16

Fermat original modular exponent bit1 squared base reduction exact quotient := div(16, 12)

Fermat original modular exponent bit1 squared base reduction exact quotient = 4/3

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(1, 12)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 12

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(16, 12)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 4

Fermat original modular exponent bit1 squared base reduction:signed integer value=16;positive modulus=12;exact quotient=4/3;floor q=1

value=q*modulus+remainder;normalized remainder=4 in[0,12)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[10, 4, 5];low bit=1

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

Fermat original modular exponent bit2 current base square := mul(4, 4)

Fermat original modular exponent bit2 current base square = 16

Fermat original modular exponent bit2 squared base reduction exact quotient := div(16, 12)

Fermat original modular exponent bit2 squared base reduction exact quotient = 4/3

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(1, 12)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 12

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(16, 12)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 4

Fermat original modular exponent bit2 squared base reduction:signed integer value=16;positive modulus=12;exact quotient=4/3;floor q=1

value=q*modulus+remainder;normalized remainder=4 in[0,12)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[4, 4, 2];low bit=0

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

Fermat original modular exponent bit3 accumulator product := mul(4, 4)

Fermat original modular exponent bit3 accumulator product = 16

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(16, 12)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 4/3

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(1, 12)

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple = 12

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(16, 12)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 4

Fermat original modular exponent bit3 accumulator reduction:signed integer value=16;positive modulus=12;exact quotient=4/3;floor q=1

value=q*modulus+remainder;normalized remainder=4 in[0,12)

Fermat original modular exponent bit3 current base square := mul(4, 4)

Fermat original modular exponent bit3 current base square = 16

Fermat original modular exponent bit3 squared base reduction exact quotient := div(16, 12)

Fermat original modular exponent bit3 squared base reduction exact quotient = 4/3

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(1, 12)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 12

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(16, 12)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 4

Fermat original modular exponent bit3 squared base reduction:signed integer value=16;positive modulus=12;exact quotient=4/3;floor q=1

value=q*modulus+remainder;normalized remainder=4 in[0,12)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[4, 4, 1];low bit=1

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

Fermat original modular exponent:remaining exponent0;final normalized accumulator=4

EXACT original base^11 modulo12=4

ACTUAL modulus=12;ORIGINAL signed base=10;prime=否;coprime=否;actual power residue=4

prime-coprime applicability=否;failed hypotheses do NOT force actual residue!=1;one successful residue does NOT prove prime

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction normalized remainder，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction floor quotient multiple，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction exact quotient，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 current base square，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 Fermat original modular exponent bit3 accumulator reduction normalized remainder，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 accumulator reduction floor quotient multiple，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 accumulator reduction exact quotient，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 accumulator product，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 Fermat original modular exponent bit2 squared base reduction normalized remainder，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction floor quotient multiple，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction exact quotient，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 current base square，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 Fermat original modular exponent bit1 squared base reduction normalized remainder，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base 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 Fermat original modular exponent bit1 squared base reduction exact quotient，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 current base square，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 Fermat original modular exponent bit1 accumulator reduction normalized remainder，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 accumulator reduction floor quotient multiple，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 accumulator reduction exact quotient，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 accumulator product，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction normalized remainder，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction floor quotient multiple，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction exact quotient，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 current base square，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original modular exponent bit0 accumulator reduction normalized remainder，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 accumulator reduction floor quotient multiple，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 accumulator reduction exact quotient，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 accumulator product，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original signed base normalized remainder，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base floor quotient multiple，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base exact quotient，不以小數近似替代。
• S003 · 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。
• S002 · 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。
• 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

(12, 10, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)

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。