Exact Division Chains, Signed Bezout Witnesses and Zero Boundaries
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1000, b=97. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1000, 97, 10, 30], [97, 30, 3, 7], [30, 7, 4, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1000, 97, 10, 30], [97, 30, 3, 7], [30, 7, 4, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
2. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1037, b=98. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1037, 98, 10, 57], [98, 57, 1, 41], [57, 41, 1, 16], [41, 16, 2, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1037, 98, 10, 57], [98, 57, 1, 41], [57, 41, 1, 16], [41, 16, 2, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
3. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1074, b=99. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1074, 99, 10, 84], [99, 84, 1, 15], [84, 15, 5, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1074, 99, 10, 84], [99, 84, 1, 15], [84, 15, 5, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
4. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1111, b=100. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1111, 100, 11, 11], [100, 11, 9, 1], [11, 1, 11, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1111, 100, 11, 11], [100, 11, 9, 1], [11, 1, 11, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
5. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1148, b=101. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1148, 101, 11, 37], [101, 37, 2, 27], [37, 27, 1, 10], [27, 10, 2, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1148, 101, 11, 37], [101, 37, 2, 27], [37, 27, 1, 10], [27, 10, 2, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
6. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1185, b=102. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1185, 102, 11, 63], [102, 63, 1, 39], [63, 39, 1, 24], [39, 24, 1, 15], [24, 15, 1, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1185, 102, 11, 63], [102, 63, 1, 39], [63, 39, 1, 24], [39, 24, 1, 15], [24, 15, 1, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
7. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1222, b=103. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1222, 103, 11, 89], [103, 89, 1, 14], [89, 14, 6, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1222, 103, 11, 89], [103, 89, 1, 14], [89, 14, 6, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
8. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1259, b=104. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1259, 104, 12, 11], [104, 11, 9, 5], [11, 5, 2, 1], [5, 1, 5, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1259, 104, 12, 11], [104, 11, 9, 5], [11, 5, 2, 1], [5, 1, 5, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
9. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1296, b=105. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1296, 105, 12, 36], [105, 36, 2, 33], [36, 33, 1, 3], [33, 3, 11, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1296, 105, 12, 36], [105, 36, 2, 33], [36, 33, 1, 3], [33, 3, 11, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
10. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1333, b=106. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1333, 106, 12, 61], [106, 61, 1, 45], [61, 45, 1, 16], [45, 16, 2, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1333, 106, 12, 61], [106, 61, 1, 45], [61, 45, 1, 16], [45, 16, 2, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
11. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1370, b=107. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1370, 107, 12, 86], [107, 86, 1, 21], [86, 21, 4, 2], [21, 2, 10, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1370, 107, 12, 86], [107, 86, 1, 21], [86, 21, 4, 2], [21, 2, 10, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
12. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1407, b=108. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1407, 108, 13, 3], [108, 3, 36, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1407, 108, 13, 3], [108, 3, 36, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
13. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1444, b=109. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1444, 109, 13, 27], [109, 27, 4, 1], [27, 1, 27, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1444, 109, 13, 27], [109, 27, 4, 1], [27, 1, 27, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
14. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1481, b=110. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1481, 110, 13, 51], [110, 51, 2, 8], [51, 8, 6, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1481, 110, 13, 51], [110, 51, 2, 8], [51, 8, 6, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
15. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1518, b=111. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1518, 111, 13, 75], [111, 75, 1, 36], [75, 36, 2, 3], [36, 3, 12, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1518, 111, 13, 75], [111, 75, 1, 36], [75, 36, 2, 3], [36, 3, 12, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
16. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1555, b=112. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1555, 112, 13, 99], [112, 99, 1, 13], [99, 13, 7, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1555, 112, 13, 99], [112, 99, 1, 13], [99, 13, 7, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
17. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1592, b=113. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
18. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1629, b=114. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1629, 114, 14, 33], [114, 33, 3, 15], [33, 15, 2, 3], [15, 3, 5, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1629, 114, 14, 33], [114, 33, 3, 15], [33, 15, 2, 3], [15, 3, 5, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
19. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1666, b=115. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1666, 115, 14, 56], [115, 56, 2, 3], [56, 3, 18, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1666, 115, 14, 56], [115, 56, 2, 3], [56, 3, 18, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
20. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1703, b=116. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1703, 116, 14, 79], [116, 79, 1, 37], [79, 37, 2, 5], [37, 5, 7, 2], [5, 2, 2, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1703, 116, 14, 79], [116, 79, 1, 37], [79, 37, 2, 5], [37, 5, 7, 2], [5, 2, 2, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
21. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1740, b=117. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1740, 117, 14, 102], [117, 102, 1, 15], [102, 15, 6, 12], [15, 12, 1, 3], [12, 3, 4, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1740, 117, 14, 102], [117, 102, 1, 15], [102, 15, 6, 12], [15, 12, 1, 3], [12, 3, 4, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
22. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1777, b=118. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1777, 118, 15, 7], [118, 7, 16, 6], [7, 6, 1, 1], [6, 1, 6, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1777, 118, 15, 7], [118, 7, 16, 6], [7, 6, 1, 1], [6, 1, 6, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
23. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1814, b=119. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1814, 119, 15, 29], [119, 29, 4, 3], [29, 3, 9, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1814, 119, 15, 29], [119, 29, 4, 3], [29, 3, 9, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
24. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1851, b=120. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1851, 120, 15, 51], [120, 51, 2, 18], [51, 18, 2, 15], [18, 15, 1, 3], [15, 3, 5, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1851, 120, 15, 51], [120, 51, 2, 18], [51, 18, 2, 15], [18, 15, 1, 3], [15, 3, 5, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
25. Exact Quotient-Remainder Steps and Strict Descent Are Both Required
Use original integers a=1888, b=121. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).
View answer and explanation
([[1888, 121, 15, 73], [121, 73, 1, 48], [73, 48, 1, 25], [48, 25, 1, 23], [25, 23, 1, 2], [23, 2, 11, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)
Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1888, 121, 15, 73], [121, 73, 1, 48], [73, 48, 1, 25], [48, 25, 1, 23], [25, 23, 1, 2], [23, 2, 11, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-division-chain · candidate
26. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1000, b=97. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
27. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1037, b=98. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
28. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1074, b=99. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
29. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1111, b=100. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
30. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1148, b=101. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
31. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1185, b=102. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
32. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1222, b=103. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
33. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1259, b=104. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
34. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1296, b=105. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
35. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1333, b=106. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
36. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1370, b=107. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
37. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1407, b=108. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
38. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1444, b=109. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
39. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1481, b=110. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
40. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1518, b=111. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
41. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1555, b=112. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
42. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1592, b=113. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
43. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1629, b=114. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
44. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1666, b=115. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
45. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1703, b=116. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
46. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1740, b=117. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
47. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1777, b=118. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
48. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1814, b=119. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
49. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1851, b=120. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
50. Gcd Preservation and Termination Have Different Obligations
Use original integers a=1888, b=121. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd_pairs, gcd, invariant, certificate).
View answer and explanation
([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)
For each row compare gcd(left,right) with gcd(right,remainder), retaining all pairs in chain order. Gcd invariance identifies the preserved value; strict nonnegative-remainder descent supplies termination separately. The exact result, retaining the requested order, is ([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-gcd-invariant · candidate
51. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=401, b=89. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, 2, -9, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, 2, -9, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
52. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-424, b=91. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -44, -205, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -44, -205, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
53. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=447, b=93. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(3, 5, -24, 3, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (3, 5, -24, 3, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
54. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-470, b=95. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(5, 1, 5, 5, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (5, 1, 5, 5, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
55. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=493, b=97. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -12, 61, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -12, 61, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
56. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-516, b=99. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(3, 14, 73, 3, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (3, 14, 73, 3, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
57. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=539, b=101. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, 3, -16, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, 3, -16, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
58. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-562, b=103. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, 46, 251, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, 46, 251, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
59. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=585, b=105. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(15, 2, -11, 15, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (15, 2, -11, 15, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
60. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-608, b=107. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -22, -125, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -22, -125, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
61. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=631, b=109. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -19, 110, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -19, 110, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
62. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-654, b=111. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(3, -9, -53, 3, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (3, -9, -53, 3, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
63. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=677, b=113. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -1, 6, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -1, 6, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
64. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-700, b=115. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(5, 11, 67, 5, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (5, 11, 67, 5, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
65. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=723, b=117. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(3, -11, 68, 3, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (3, -11, 68, 3, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
66. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-746, b=119. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, 26, 163, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, 26, 163, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
67. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=769, b=121. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -45, 286, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -45, 286, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
68. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-792, b=123. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(3, -16, -103, 3, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (3, -16, -103, 3, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
69. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=815, b=125. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(5, 2, -13, 5, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (5, 2, -13, 5, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
70. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-838, b=127. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, 5, 33, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, 5, 33, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
71. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=861, b=129. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(3, 3, -20, 3, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (3, 3, -20, 3, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
72. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-884, b=131. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, 4, 27, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, 4, 27, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
73. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=907, b=133. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -61, 416, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -61, 416, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
74. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=-930, b=135. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(15, 1, 7, 15, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (15, 1, 7, 15, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
75. Restore Input Signs in the Extended Euclidean Witness
Use original integers a=953, b=137. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).
View answer and explanation
(1, -23, 160, 1, Yes, extended_euclidean_back_substitution)
Use extended Euclidean back-substitution to obtain the source deterministic coefficients x,y, then verify a*x+b*y equals the nonnegative gcd. Exclude (0,0) here. A correct gcd value alone does not determine valid Bezout coefficients. The exact result, retaining the requested order, is (1, -23, 160, 1, Yes, extended_euclidean_back_substitution).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-bezout-certificate · candidate
76. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 0, 0, 1, 0, 0, gcd_0_0_equals_0)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 0, 0, 1, 0, 0, gcd_0_0_equals_0).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
77. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=2. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 2, 2, 0, 1, 2, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 2, 2, 0, 1, 2, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
78. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-3, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-3, 0, 3, -1, 0, 3, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
79. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=4. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 4, 4, 0, 1, 4, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 4, 4, 0, 1, 4, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
80. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-5, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-5, 0, 5, -1, 0, 5, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-5, 0, 5, -1, 0, 5, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
81. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=6. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 6, 6, 0, 1, 6, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 6, 6, 0, 1, 6, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
82. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-7, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-7, 0, 7, -1, 0, 7, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-7, 0, 7, -1, 0, 7, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
83. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=8. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 8, 8, 0, 1, 8, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 8, 8, 0, 1, 8, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
84. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-9, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-9, 0, 9, -1, 0, 9, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-9, 0, 9, -1, 0, 9, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
85. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=10. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 10, 10, 0, 1, 10, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 10, 10, 0, 1, 10, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
86. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-11, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-11, 0, 11, -1, 0, 11, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-11, 0, 11, -1, 0, 11, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
87. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=12. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 12, 12, 0, 1, 12, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 12, 12, 0, 1, 12, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
88. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-13, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-13, 0, 13, -1, 0, 13, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-13, 0, 13, -1, 0, 13, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
89. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=14. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 14, 14, 0, 1, 14, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 14, 14, 0, 1, 14, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
90. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-15, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-15, 0, 15, -1, 0, 15, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-15, 0, 15, -1, 0, 15, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
91. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=16. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 16, 16, 0, 1, 16, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 16, 16, 0, 1, 16, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
92. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-17, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-17, 0, 17, -1, 0, 17, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-17, 0, 17, -1, 0, 17, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
93. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=18. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 18, 18, 0, 1, 18, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 18, 18, 0, 1, 18, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
94. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-19, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-19, 0, 19, -1, 0, 19, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-19, 0, 19, -1, 0, 19, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
95. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=20. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 20, 20, 0, 1, 20, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 20, 20, 0, 1, 20, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
96. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-21, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-21, 0, 21, -1, 0, 21, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-21, 0, 21, -1, 0, 21, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
97. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=22. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 22, 22, 0, 1, 22, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 22, 22, 0, 1, 22, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
98. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-23, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-23, 0, 23, -1, 0, 23, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-23, 0, 23, -1, 0, 23, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
99. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=0, b=24. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(0, 24, 24, 0, 1, 24, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (0, 24, 24, 0, 1, 24, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
100. Zero Inputs Have an Explicit Gcd Convention without Division by Zero
Use original integers a=-25, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).
View answer and explanation
(-25, 0, 25, -1, 0, 25, no_division_by_zero_executed)
Require at least one zero input. Give original signed a,b, the nonnegative gcd, deterministic x,y and a*x+b*y. At (0,0) retain source coefficients (1,0) and gcd0; otherwise use the no-division-by-zero convention. The exact result, retaining the requested order, is (-25, 0, 25, -1, 0, 25, no_division_by_zero_executed).
Reviewed step-by-step derivation
Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate
Canonical question JSONL · English JSONL · Validation report