# Exact Division Chains, Signed Bezout Witnesses and Zero Boundaries

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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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0100/)

Euclidean algorithm · qf-mod36-euclid-zero-input-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations/validation-report.json)