第 18815 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":723,"b":117}。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":723,"b":117}。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=723,b=117

initial nonnegative chain values=[723, 117];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(723, 117)

Euclid step0 exact ratio = 241/39

Euclid step0 quotient times divisor := mul(6, 117)

Euclid step0 quotient times divisor = 702

Euclid step0 remainder := sub(723, 702)

Euclid step0 remainder = 21

step=0;left=723;positive right=117;exact ratio=241/39;floor quotient=6

left=right*q+r with r=21 in[0,117);gcd pair=[3, 3]

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=21

Euclid step1 exact ratio := div(117, 21)

Euclid step1 exact ratio = 39/7

Euclid step1 quotient times divisor := mul(5, 21)

Euclid step1 quotient times divisor = 105

Euclid step1 remainder := sub(117, 105)

Euclid step1 remainder = 12

step=1;left=117;positive right=21;exact ratio=39/7;floor quotient=5

left=right*q+r with r=12 in[0,21);gcd pair=[3, 3]

Euclid step1 first coefficient multiple := mul(5, 1)

Euclid step1 first coefficient multiple = 5

Euclid step1 first coefficient difference := sub(0, 5)

Euclid step1 first coefficient difference = -5

Euclid step1 second coefficient multiple := mul(5, -6)

Euclid step1 second coefficient multiple = -30

Euclid step1 second coefficient difference := sub(1, -30)

Euclid step1 second coefficient difference = 31

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=5

next remainder coefficient pair=[-5, 31];abs(a)*s_next+abs(b)*t_next=12

Euclid step2 exact ratio := div(21, 12)

Euclid step2 exact ratio = 7/4

Euclid step2 quotient times divisor := mul(1, 12)

Euclid step2 quotient times divisor = 12

Euclid step2 remainder := sub(21, 12)

Euclid step2 remainder = 9

step=2;left=21;positive right=12;exact ratio=7/4;floor quotient=1

left=right*q+r with r=9 in[0,12);gcd pair=[3, 3]

Euclid step2 first coefficient multiple := mul(1, -5)

Euclid step2 first coefficient multiple = -5

Euclid step2 first coefficient difference := sub(1, -5)

Euclid step2 first coefficient difference = 6

Euclid step2 second coefficient multiple := mul(1, 31)

Euclid step2 second coefficient multiple = 31

Euclid step2 second coefficient difference := sub(-6, 31)

Euclid step2 second coefficient difference = -37

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-5, 31]];same quotient=1

next remainder coefficient pair=[6, -37];abs(a)*s_next+abs(b)*t_next=9

Euclid step3 exact ratio := div(12, 9)

Euclid step3 exact ratio = 4/3

Euclid step3 quotient times divisor := mul(1, 9)

Euclid step3 quotient times divisor = 9

Euclid step3 remainder := sub(12, 9)

Euclid step3 remainder = 3

step=3;left=12;positive right=9;exact ratio=4/3;floor quotient=1

left=right*q+r with r=3 in[0,9);gcd pair=[3, 3]

Euclid step3 first coefficient multiple := mul(1, 6)

Euclid step3 first coefficient multiple = 6

Euclid step3 first coefficient difference := sub(-5, 6)

Euclid step3 first coefficient difference = -11

Euclid step3 second coefficient multiple := mul(1, -37)

Euclid step3 second coefficient multiple = -37

Euclid step3 second coefficient difference := sub(31, -37)

Euclid step3 second coefficient difference = 68

step=3;two original-absolute-input coefficient pairs=[[-5, 31], [6, -37]];same quotient=1

next remainder coefficient pair=[-11, 68];abs(a)*s_next+abs(b)*t_next=3

Euclid step4 exact ratio := div(9, 3)

Euclid step4 exact ratio = 3

Euclid step4 quotient times divisor := mul(3, 3)

Euclid step4 quotient times divisor = 9

Euclid step4 remainder := sub(9, 9)

Euclid step4 remainder = 0

step=4;left=9;positive right=3;exact ratio=3;floor quotient=3

left=right*q+r with r=0 in[0,3);gcd pair=[3, 3]

Euclid step4 first coefficient multiple := mul(3, -11)

Euclid step4 first coefficient multiple = -33

Euclid step4 first coefficient difference := sub(6, -33)

Euclid step4 first coefficient difference = 39

Euclid step4 second coefficient multiple := mul(3, 68)

Euclid step4 second coefficient multiple = 204

Euclid step4 second coefficient difference := sub(-37, 204)

Euclid step4 second coefficient difference = -241

step=4;two original-absolute-input coefficient pairs=[[6, -37], [-11, 68]];same quotient=3

next remainder coefficient pair=[39, -241];abs(a)*s_next+abs(b)*t_next=0

complete division rows=[[723, 117, 6, 21], [117, 21, 5, 12], [21, 12, 1, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal pair=[3, 0]

gcd(a,b)=gcd(3,0)=3;all divisions used positive divisors;nonnegative gcd convention

Euclid restore original a sign in coefficient := mul(1, -11)

Euclid restore original a sign in coefficient = -11

Euclid restore original b sign in coefficient := mul(1, 68)

Euclid restore original b sign in coefficient = 68

Euclid ORIGINAL signed a times x := mul(723, -11)

Euclid ORIGINAL signed a times x = -7953

Euclid ORIGINAL signed b times y := mul(117, 68)

Euclid ORIGINAL signed b times y = 7956

Euclid final signed Bezout linear combination := sum(-7953, 7956)

Euclid final signed Bezout linear combination = 3

original a=723,b=117;deterministic absolute-chain coefficients=[-11, 68]

sign-restored source coefficients x=-11,y=68;a*x+b*y=3=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

(3, -11, 68, 3, 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。