第 18820 題：Euclidean演算法

原題與已知條件

exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={"a":-838,"b":127}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={"a":-838,"b":127}。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-euclidean-algorithm

逐步推導

source operation=euclid-bezout-certificate;original snapshot and mathematics retained

explicit ordered source output fields=[gcd, x, y, linear_combination, equal, certificate]

original signed inputs a=-838,b=127

initial nonnegative chain values=[838, 127];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(838, 127)

Euclid step0 exact ratio = 838/127

Euclid step0 quotient times divisor := mul(6, 127)

Euclid step0 quotient times divisor = 762

Euclid step0 remainder := sub(838, 762)

Euclid step0 remainder = 76

step=0;left=838;positive right=127;exact ratio=838/127;floor quotient=6

left=right*q+r with r=76 in[0,127);gcd pair=[1, 1]

Euclid step0 first coefficient multiple := mul(6, 0)

Euclid step0 first coefficient multiple = 0

Euclid step0 first coefficient difference := sub(1, 0)

Euclid step0 first coefficient difference = 1

Euclid step0 second coefficient multiple := mul(6, 1)

Euclid step0 second coefficient multiple = 6

Euclid step0 second coefficient difference := sub(0, 6)

Euclid step0 second coefficient difference = -6

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=6

next remainder coefficient pair=[1, -6];abs(a)*s_next+abs(b)*t_next=76

Euclid step1 exact ratio := div(127, 76)

Euclid step1 exact ratio = 127/76

Euclid step1 quotient times divisor := mul(1, 76)

Euclid step1 quotient times divisor = 76

Euclid step1 remainder := sub(127, 76)

Euclid step1 remainder = 51

step=1;left=127;positive right=76;exact ratio=127/76;floor quotient=1

left=right*q+r with r=51 in[0,76);gcd pair=[1, 1]

Euclid step1 first coefficient multiple := mul(1, 1)

Euclid step1 first coefficient multiple = 1

Euclid step1 first coefficient difference := sub(0, 1)

Euclid step1 first coefficient difference = -1

Euclid step1 second coefficient multiple := mul(1, -6)

Euclid step1 second coefficient multiple = -6

Euclid step1 second coefficient difference := sub(1, -6)

Euclid step1 second coefficient difference = 7

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=1

next remainder coefficient pair=[-1, 7];abs(a)*s_next+abs(b)*t_next=51

Euclid step2 exact ratio := div(76, 51)

Euclid step2 exact ratio = 76/51

Euclid step2 quotient times divisor := mul(1, 51)

Euclid step2 quotient times divisor = 51

Euclid step2 remainder := sub(76, 51)

Euclid step2 remainder = 25

step=2;left=76;positive right=51;exact ratio=76/51;floor quotient=1

left=right*q+r with r=25 in[0,51);gcd pair=[1, 1]

Euclid step2 first coefficient multiple := mul(1, -1)

Euclid step2 first coefficient multiple = -1

Euclid step2 first coefficient difference := sub(1, -1)

Euclid step2 first coefficient difference = 2

Euclid step2 second coefficient multiple := mul(1, 7)

Euclid step2 second coefficient multiple = 7

Euclid step2 second coefficient difference := sub(-6, 7)

Euclid step2 second coefficient difference = -13

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-1, 7]];same quotient=1

next remainder coefficient pair=[2, -13];abs(a)*s_next+abs(b)*t_next=25

Euclid step3 exact ratio := div(51, 25)

Euclid step3 exact ratio = 51/25

Euclid step3 quotient times divisor := mul(2, 25)

Euclid step3 quotient times divisor = 50

Euclid step3 remainder := sub(51, 50)

Euclid step3 remainder = 1

step=3;left=51;positive right=25;exact ratio=51/25;floor quotient=2

left=right*q+r with r=1 in[0,25);gcd pair=[1, 1]

Euclid step3 first coefficient multiple := mul(2, 2)

Euclid step3 first coefficient multiple = 4

Euclid step3 first coefficient difference := sub(-1, 4)

Euclid step3 first coefficient difference = -5

Euclid step3 second coefficient multiple := mul(2, -13)

Euclid step3 second coefficient multiple = -26

Euclid step3 second coefficient difference := sub(7, -26)

Euclid step3 second coefficient difference = 33

step=3;two original-absolute-input coefficient pairs=[[-1, 7], [2, -13]];same quotient=2

next remainder coefficient pair=[-5, 33];abs(a)*s_next+abs(b)*t_next=1

Euclid step4 exact ratio := div(25, 1)

Euclid step4 exact ratio = 25

Euclid step4 quotient times divisor := mul(25, 1)

Euclid step4 quotient times divisor = 25

Euclid step4 remainder := sub(25, 25)

Euclid step4 remainder = 0

step=4;left=25;positive right=1;exact ratio=25;floor quotient=25

left=right*q+r with r=0 in[0,1);gcd pair=[1, 1]

Euclid step4 first coefficient multiple := mul(25, -5)

Euclid step4 first coefficient multiple = -125

Euclid step4 first coefficient difference := sub(2, -125)

Euclid step4 first coefficient difference = 127

Euclid step4 second coefficient multiple := mul(25, 33)

Euclid step4 second coefficient multiple = 825

Euclid step4 second coefficient difference := sub(-13, 825)

Euclid step4 second coefficient difference = -838

step=4;two original-absolute-input coefficient pairs=[[2, -13], [-5, 33]];same quotient=25

next remainder coefficient pair=[127, -838];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[838, 127, 6, 76], [127, 76, 1, 51], [76, 51, 1, 25], [51, 25, 2, 1], [25, 1, 25, 0]];terminal pair=[1, 0]

gcd(a,b)=gcd(1,0)=1;all divisions used positive divisors;nonnegative gcd convention

Euclid restore original a sign in coefficient := mul(-1, -5)

Euclid restore original a sign in coefficient = 5

Euclid restore original b sign in coefficient := mul(1, 33)

Euclid restore original b sign in coefficient = 33

Euclid ORIGINAL signed a times x := mul(-838, 5)

Euclid ORIGINAL signed a times x = -4190

Euclid ORIGINAL signed b times y := mul(127, 33)

Euclid ORIGINAL signed b times y = 4191

Euclid final signed Bezout linear combination := sum(-4190, 4191)

Euclid final signed Bezout linear combination = 1

original a=-838,b=127;deterministic absolute-chain coefficients=[-5, 33]

sign-restored source coefficients x=5,y=33;a*x+b*y=1=nonnegative gcd

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 — 以精確有理數重算 Euclid final signed Bezout linear combination，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Euclid ORIGINAL signed b times y，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Euclid ORIGINAL signed a times x，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Euclid restore original b sign in coefficient，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Euclid restore original a sign in coefficient，不以小數近似替代。
• 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 — 以精確有理數重算 Euclid step4 second coefficient difference，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Euclid step4 second coefficient multiple，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Euclid step4 first coefficient difference，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Euclid step4 first coefficient multiple，不以小數近似替代。
• S042 · 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。
• S041 · premise_and_result_audit — 以精確有理數重算 Euclid step4 remainder，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euclid step4 quotient times divisor，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euclid step4 exact ratio，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 — 以精確有理數重算 Euclid step3 second coefficient difference，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Euclid step3 second coefficient multiple，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Euclid step3 first coefficient difference，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euclid step3 first coefficient multiple，不以小數近似替代。
• S033 · 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。
• S032 · premise_and_result_audit — 以精確有理數重算 Euclid step3 remainder，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Euclid step3 quotient times divisor，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Euclid step3 exact ratio，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 Euclid step2 second coefficient difference，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Euclid step2 second coefficient multiple，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Euclid step2 first coefficient difference，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Euclid step2 first coefficient multiple，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 Euclid step2 remainder，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Euclid step2 quotient times divisor，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Euclid step2 exact ratio，不以小數近似替代。
• 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 — 以精確有理數重算 Euclid step1 second coefficient difference，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Euclid step1 second coefficient multiple，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Euclid step1 first coefficient difference，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Euclid step1 first coefficient multiple，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 — 以精確有理數重算 Euclid step1 remainder，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Euclid step1 quotient times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Euclid step1 exact ratio，不以小數近似替代。
• 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 — 以精確有理數重算 Euclid step0 second coefficient difference，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Euclid step0 second coefficient multiple，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Euclid step0 first coefficient difference，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Euclid step0 first coefficient multiple，不以小數近似替代。
• 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 — 以精確有理數重算 Euclid step0 remainder，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Euclid step0 quotient times divisor，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Euclid step0 exact ratio，不以小數近似替代。
• 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

(1, 5, 33, 1, 是, extended_euclidean_back_substitution)

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。