{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0001","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc0c4d44e8b06228cd845b19daefee198e8f748ccadc707088f141c4ab12a400","family_id":"qf-mod36-euclid-division-chain","seed":202611053001,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1000,\"b\":97}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0002","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6667feb7260139c0816ce6c698c73133c56c71abdff4be9d32cd15a8c05251c0","family_id":"qf-mod36-euclid-division-chain","seed":202611053002,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1037,\"b\":98}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0003","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44601196e567d40ad4d45e746078462361db2be7e08ae579362bbd4207a4baa0","family_id":"qf-mod36-euclid-division-chain","seed":202611053003,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1074,\"b\":99}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0004","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b048e409ec7113154c4aad94c993e9022012704f2ed00c60a26462bf1d5eca46","family_id":"qf-mod36-euclid-division-chain","seed":202611053004,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1111,\"b\":100}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1111, 100, 11, 11], [100, 11, 9, 1], [11, 1, 11, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0005","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cb20bac99ba1cabdcfad3b260f663534243314b3e4c1b161b3699901344a8b6","family_id":"qf-mod36-euclid-division-chain","seed":202611053005,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1148,\"b\":101}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0006","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b1e038f4f34fe29513450bbf9dbd3915743c0e1c448c9c50004e411807fb4b9","family_id":"qf-mod36-euclid-division-chain","seed":202611053006,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1185,\"b\":102}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0007","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"005edecd009bd9b6b28b8acbe911f015060945f6c015f82d5829ac45a0799d7d","family_id":"qf-mod36-euclid-division-chain","seed":202611053007,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1222,\"b\":103}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0008","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f40271bb5c88e8620ab0185425d62233827bda9cbdc25d6acd4ef53f6bca437e","family_id":"qf-mod36-euclid-division-chain","seed":202611053008,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1259,\"b\":104}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1259, 104, 12, 11], [104, 11, 9, 5], [11, 5, 2, 1], [5, 1, 5, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0009","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f778efb19e7fcac179ab3bffa0d7bf28772cd05b0702ef73e74ab628936b613","family_id":"qf-mod36-euclid-division-chain","seed":202611053009,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1296,\"b\":105}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1296, 105, 12, 36], [105, 36, 2, 33], [36, 33, 1, 3], [33, 3, 11, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0010","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02105ab7c24874c154bc1036def8c758ef543d89cb759f2b2dc23c9d3af38163","family_id":"qf-mod36-euclid-division-chain","seed":202611053010,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1333,\"b\":106}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0011","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f7c1c6a75fea1c88f16645430d02fac7bf6b15d085ded528e60527f1614e905","family_id":"qf-mod36-euclid-division-chain","seed":202611053011,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1370,\"b\":107}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0012","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"687a8a1755bef8da29772abe90d7afebcc93f3f3f60e4b879d1e84e09df41e19","family_id":"qf-mod36-euclid-division-chain","seed":202611053012,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1407,\"b\":108}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1407, 108, 13, 3], [108, 3, 36, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0013","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cf95d413dc0781b969696bfb8bd39302ba7c57d8203972de4ad668d48c85100","family_id":"qf-mod36-euclid-division-chain","seed":202611053013,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1444,\"b\":109}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1444, 109, 13, 27], [109, 27, 4, 1], [27, 1, 27, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0014","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25192d99fdf71a32f2535196f8a6dc8db96085f12da2e828ebd670eee1cc8df5","family_id":"qf-mod36-euclid-division-chain","seed":202611053014,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1481,\"b\":110}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0015","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1154f1e7b5ff08f646ea14ea6180f7c5c0d304871c4ea34687416d25ddbc9fe2","family_id":"qf-mod36-euclid-division-chain","seed":202611053015,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1518,\"b\":111}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1518, 111, 13, 75], [111, 75, 1, 36], [75, 36, 2, 3], [36, 3, 12, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0016","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4307bd49f0f63ec535163d5e85cd08341507b71362c93dc0ca548cf9c42196f6","family_id":"qf-mod36-euclid-division-chain","seed":202611053016,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1555,\"b\":112}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0017","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58ef397a5fe4396edb56675fc869c95ad6d5426e9ce8eb0696de7abdee6bd929","family_id":"qf-mod36-euclid-division-chain","seed":202611053017,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1592,\"b\":113}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0018","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69726fe6faaa99ea056b97840de8a571dcb0fd2e13f31198d55d36b8ee2f0463","family_id":"qf-mod36-euclid-division-chain","seed":202611053018,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1629,\"b\":114}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1629, 114, 14, 33], [114, 33, 3, 15], [33, 15, 2, 3], [15, 3, 5, 0]], 3, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0019","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02d979762c3f60d46a2eb8c15216e38ae36ec8ab2c3e6011bd621f9bf683687c","family_id":"qf-mod36-euclid-division-chain","seed":202611053019,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1666,\"b\":115}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0020","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a931fc1128f47dc7e95b9c4c9c47c1da700b6a9aab5acf3665658c77b000694","family_id":"qf-mod36-euclid-division-chain","seed":202611053020,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1703,\"b\":116}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0021","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"858a8187d0d1ef6f7c66ec5aed6d0386494aa29dcb0dead7ba75e5c4575cd03d","family_id":"qf-mod36-euclid-division-chain","seed":202611053021,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1740,\"b\":117}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0022","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3661c216802a569f07f33e8cf047afa8f9a34fe144f7428343ebec3f0366bf7","family_id":"qf-mod36-euclid-division-chain","seed":202611053022,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1777,\"b\":118}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[1777, 118, 15, 7], [118, 7, 16, 6], [7, 6, 1, 1], [6, 1, 6, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0023","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec5dae3999f93401fb938d2a91d2857fc6c4b130257e9d3702592705bab7e8c5","family_id":"qf-mod36-euclid-division-chain","seed":202611053023,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1814,\"b\":119}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0024","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49141f5baf2da05e8ace494b41d2408bba3a8ff62656f8ee7d7f58069b4fc8a1","family_id":"qf-mod36-euclid-division-chain","seed":202611053024,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1851,\"b\":120}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0025","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2edfaf55f4c361149836ed5c31e9c4b33ef2464921a77d6fecde8839ec3616ac","family_id":"qf-mod36-euclid-division-chain","seed":202611053025,"semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1888,\"b\":121}"},"fields":{"title":"Exact Quotient-Remainder Steps and Strict Descent Are Both Required","stem":"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).","answer":{"kind":"expression","value":"([[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)","explanation":"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)."},"skills":["Exact Quotient-Remainder Steps and Strict Descent Are Both Required"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0026","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56b3c4c4025fa5bcf9f0c77541bc362e87faa93e91d0991fbef498d1ac0777ca","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053026,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1000,\"b\":97}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0027","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edd17f6b58fadb47e711f5ad2707f83869479b9500f0159aca656f754dca9601","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053027,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1037,\"b\":98}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0028","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f5af1832d38220e62683e05b40102abc28799777c973f254218194a7b4e3e04","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053028,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1074,\"b\":99}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0029","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"790dfd333308c6ddc425ae5fc6292d04021cd5623cb440705fbe9f140f33e731","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053029,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1111,\"b\":100}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0030","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5823001ae6597f5d032220d441dce0956a6e98db4a90bdad4a9b0d6480eab259","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053030,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1148,\"b\":101}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0031","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e39c5bc8e855cfa30e1d9855176420fec0bf4bb06e2d105bf1914ccb7be91e66","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053031,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1185,\"b\":102}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0032","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a05a7693f5a0b7468c5d3e08af1ceb755b28512118b739a6e8e08198c8690f74","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053032,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1222,\"b\":103}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0033","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"665ea79fcbd62aaca48554ba8cd369489bebb41f22ca36c173f69b8395ac5f5c","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053033,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1259,\"b\":104}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0034","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e608a1532ab5653aee223e13b6377c5a08a525c8bc6052de4eb676ce3b5879ca","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053034,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1296,\"b\":105}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0035","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"627e4be1dd53a4d71fc656e96f30c83d5b94defd29badf08a032bab2a003ca54","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053035,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1333,\"b\":106}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0036","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4645912c44e785261d79b2b5d766bb351f66d6f27c2f3634d314e5c3d5527b43","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053036,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1370,\"b\":107}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0037","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c8cdd1127c64c460ad6ceed177f1226252d4c670b3ffd81885bfcc53e61e7fe","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053037,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1407,\"b\":108}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0038","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e548d78719c96fa6e291547d30d27bfccd9af3c2876567d8bcfc0c56cca006ef","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053038,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1444,\"b\":109}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0039","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0127f0aa06de8e647660f79f99a24834c5d08b33981d618e7795f6604504ebd2","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053039,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1481,\"b\":110}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0040","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d835c054d413bbb36c3f416df44655734f3693f36645fe8146ca3651c20ba034","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053040,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1518,\"b\":111}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0041","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cf747431a3f6b939958b2c81a13ff09ca37a2c5ceb7f2d25247619ee1a12fb6","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053041,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1555,\"b\":112}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0042","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f83b877e0e82d494e773503294cf1b6d897519dbad98a058f39a47bf6864f18","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053042,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1592,\"b\":113}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0043","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"098caf6e9b04c3d48799538a560e48b160afc8619b4c696f5e6ddae9432b8b87","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053043,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1629,\"b\":114}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0044","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f6d621997999cc1d78befc816c2243c631b7f0ba2de88be8dd8b2fa8759ee4d","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053044,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1666,\"b\":115}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0045","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f669ee5331c2ea97ae31045cd79913409027e57d9377511c9869eaa8935d1be","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053045,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1703,\"b\":116}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0046","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e3cf583901f497d6608dfc29c7367738a0e0ca1a5ef7557b2d09c4606b395cc","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053046,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1740,\"b\":117}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0047","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b605f010d3fbd58dd7e2058ad2b12ae9ff91b77cf673342f32fc380177baec22","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053047,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1777,\"b\":118}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0048","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e6b07479a60f38faf4c1c544c899f333b492926f167fbbbc326d1410e06ab2c","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053048,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1814,\"b\":119}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0049","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3b40095f54a8a61071471fd0402c014b25f51d2589bd1c35c697209095f0d64","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053049,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1851,\"b\":120}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0050","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ef62f20f96d323948ca772c532ec482f3e0d31f92b66b07d71c8429b58ccd00","family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053050,"semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1888,\"b\":121}"},"fields":{"title":"Gcd Preservation and Termination Have Different Obligations","stem":"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).","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, Yes, gcd_invariant_separate_from_termination)","explanation":"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)."},"skills":["Gcd Preservation and Termination Have Different Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0051","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5d3f14b57c0585daeeb692e0bf69eba0b5fbaddeda864f92291c0830d5f4613","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053051,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":401,\"b\":89}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, 2, -9, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0052","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5333fd988f1a9e57d4650dac708a76f21fe8855f91afcba57ba570b1c3758598","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053052,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-424,\"b\":91}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -44, -205, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0053","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"444e533dc7817b6f8128ab803c980b9b6a29fcd77e253d0a8ed5ac40bba03496","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053053,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":447,\"b\":93}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(3, 5, -24, 3, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0054","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"446db7a15c905e0aeeb1474957e112f4ed48d1d5d60d8339ec2ff3e8467cad3e","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053054,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-470,\"b\":95}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(5, 1, 5, 5, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0055","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccc1a639a68642e5c6f30794fba149fef689192ff9c696c1788707a62f70ade4","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053055,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":493,\"b\":97}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -12, 61, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0056","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e348f3e427511e964aab2f8548695384eb85ad6f60a2b6c9abfb024d0a7b62a9","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053056,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-516,\"b\":99}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(3, 14, 73, 3, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0057","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75de2a02305b1f69008b465319344b16b5fdda3d31b41c6f5a62204587d0dff6","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053057,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":539,\"b\":101}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, 3, -16, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0058","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e81cf0988941e71382e6f43d3a3d7fc5f328dab62999f4c6202e3aaf93c159c5","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053058,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-562,\"b\":103}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, 46, 251, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0059","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04cf1364f0f72a31481beb91a0ef9ad00d6111f31e5cb9ecc56eeaf0cf3e8c7f","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053059,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":585,\"b\":105}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(15, 2, -11, 15, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0060","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f2e9b3d5d82d39a53516b0dcc555db733a0cebca98886f645e17ed5b1c7be84","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053060,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-608,\"b\":107}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -22, -125, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0061","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96f3e1c67742cf1994dbf1adaacb9dda0453994a4076fa009b0cc637db50362","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053061,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":631,\"b\":109}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -19, 110, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0062","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaca40c6ddf85d1dacc664c83875f71723512e5d96e14d3edae59e29662957da","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053062,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-654,\"b\":111}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(3, -9, -53, 3, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0063","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b51a8ed35313c46354ddc85dac0d9d3799e44ed5d1deefbe885b634a49ab7c1","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053063,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":677,\"b\":113}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -1, 6, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0064","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef4e62f7cd50f6bb87d0596eac9f3dce67c4a3d01162a375b42cd134568464b","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053064,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-700,\"b\":115}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(5, 11, 67, 5, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0065","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84d1f0d99b51b6edd5a766047ad02c71ed8f50ff9546d28780ec34ecb80f0d52","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053065,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":723,\"b\":117}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(3, -11, 68, 3, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0066","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f53f22f10786549d401d1bbeadecc6330f27a9c41366cb9a3662e5867c69a8","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053066,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-746,\"b\":119}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, 26, 163, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0067","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efba7871914ca6ea4939e9f24ac0ac1a2d03b2398ff003716254099c7df47f8d","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053067,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":769,\"b\":121}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -45, 286, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0068","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"486ee6741582054eaf2edbd35a8dc2cc69ea658a42de978a1c464470fb5f1014","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053068,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-792,\"b\":123}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(3, -16, -103, 3, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0069","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86f7e8129bf3198d333900016ab261a5796f2834d61600b2421c1c643c7dbf85","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053069,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":815,\"b\":125}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(5, 2, -13, 5, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0070","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4aa4f36d297cfaea1b2fedad772f4d72e4af5e56317df25cfb6b0412669808d","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053070,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-838,\"b\":127}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, 5, 33, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0071","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af1e457466963a3d075dbd8f0570a8bba67797604eada12517e2968720442ec5","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053071,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":861,\"b\":129}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(3, 3, -20, 3, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0072","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fbec1fbfdff251e91fb8d183247254cf66b8c187b0d36cad5566721650ac1d1","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053072,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-884,\"b\":131}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, 4, 27, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0073","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fa712447fd4558f84450740d88d8b646b02b6a4ac1726495e1986d1deed1b0f","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053073,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":907,\"b\":133}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -61, 416, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0074","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"180ab617ab88d844b5376674c99acea69340e27667abfa72f88db16e3407d5b9","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053074,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-930,\"b\":135}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(15, 1, 7, 15, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0075","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf0305930797fdfa36767fed83232809c5b8f138125bd40b196289ce8d4b694b","family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053075,"semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":953,\"b\":137}"},"fields":{"title":"Restore Input Signs in the Extended Euclidean Witness","stem":"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).","answer":{"kind":"expression","value":"(1, -23, 160, 1, Yes, extended_euclidean_back_substitution)","explanation":"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)."},"skills":["Restore Input Signs in the Extended Euclidean Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0076","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15fa3dbb2eb4a4d563d66560fdd8ff96011098fb3ed689c4293568759569bd00","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053076,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 0, 0, 1, 0, 0, gcd_0_0_equals_0)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0077","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccbb216c9fcc8fb2331b9fa0ba353b63bd481f5bd6b00fb336c8442382178972","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053077,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":2}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 2, 2, 0, 1, 2, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0078","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7001567d6efc2d749582be61bb5345f9e74273fd346e17915bfae267c04094d","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053078,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-3,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0079","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d201f91ed5394cc023d98bbb819748eff06b353fb3f82cf7e7ba8a395bfced45","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053079,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":4}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 4, 4, 0, 1, 4, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0080","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc02027cf926230f7e562bab123712ea7f6d31a45ecace40d1c533e765883830","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053080,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-5,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-5, 0, 5, -1, 0, 5, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0081","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df0eeb6354cfdf474a7bbb2ecc4c6f283d037bb3849b5d2cf25ba1526918b5a","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053081,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":6}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 6, 6, 0, 1, 6, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0082","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be03d3321aa358b44cf0e91fb6e290b735720afdd0bcbd271a3c4d7257c98a39","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053082,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-7,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-7, 0, 7, -1, 0, 7, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0083","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2d554e5c1dd30a40b9823c44b7a65b9a5f02fc59b75ac7b041414aac63ccb83","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053083,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":8}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 8, 8, 0, 1, 8, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0084","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0e2bac0fa4d89e194ef20932948477fbf66fbeadf9e6eb8503bbc737b8f2d7f","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053084,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-9,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-9, 0, 9, -1, 0, 9, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0085","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ee4fc96da8e2e9b566576eb3a085265c481cb78fda952c13b9da1b3dbd676c","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053085,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":10}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 10, 10, 0, 1, 10, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0086","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01af221cc8b024c58e164c139e01532d8d2cd8bc64b2e0873a48297a4ccc493c","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053086,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-11,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-11, 0, 11, -1, 0, 11, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0087","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1e6a507361cfd20f87a61b107a65bb6390042390315b642ecebf391fa55b63b","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053087,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":12}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 12, 12, 0, 1, 12, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0088","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b300f27d504a42251adeeb060b05cb679c62c4519572ef452b0df9a6b48d1ee0","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053088,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-13,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-13, 0, 13, -1, 0, 13, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0089","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbb8c59da4ebe6a5f9b349746097e04977f35ec1ed248ec3d0adc9429529055c","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053089,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":14}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 14, 14, 0, 1, 14, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0090","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ecca0c045eacb96e437d9a3daf9cb20ff3bdeefdfd576c534db4e1aefb00748","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053090,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-15,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-15, 0, 15, -1, 0, 15, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0091","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"626fb0984488b8c05abaa5a0218962e1c516a3463635d8adc0304f963d4b171c","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053091,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":16}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 16, 16, 0, 1, 16, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0092","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8357dd1a47e187a14c77de482922dee9df5b93abd19d32f12280bbfe55304ee","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053092,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-17,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-17, 0, 17, -1, 0, 17, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0093","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17ffdad6d8764f1ebf35f5c58db17ef2e73830f98af5110b3219f74c1aeea7bf","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053093,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":18}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 18, 18, 0, 1, 18, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0094","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dab1397a0a44354e7ddd8b2945c87008e783e9ec38ae47025c5e2361d7578b1","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053094,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-19,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-19, 0, 19, -1, 0, 19, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0095","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b1b96439e83748c1a6716bb43638f7951e910e8e270bb6a8722cdff1e278c2d","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053095,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":20}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 20, 20, 0, 1, 20, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0096","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93677233d1c3d991a4acd712f6c70fc11ba28808d47b93e6cb2e977f3e095a6f","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053096,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-21,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-21, 0, 21, -1, 0, 21, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0097","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f14ec390b7ba2df4e1a667e0dd04609ac031426c2ae7f9118f1f6103a0b39834","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053097,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":22}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 22, 22, 0, 1, 22, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0098","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d33d35253f18e0589e7657b3a4a373f62cf97d6bb2f26a357c955ba792a746d","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053098,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-23,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-23, 0, 23, -1, 0, 23, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0099","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc45a31e9b2d54e345adbebbc3cadb3ca4358aa03fbf78cc69ec96e6229c2182","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053099,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":24}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(0, 24, 24, 0, 1, 24, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0100","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdcfd8e8487fe78d5fd2740fafbf052d39dd675b7b43ac8943750b77308d852d","family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053100,"semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-25,\"b\":0}"},"fields":{"title":"Zero Inputs Have an Explicit Gcd Convention without Division by Zero","stem":"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).","answer":{"kind":"expression","value":"(-25, 0, 25, -1, 0, 25, no_division_by_zero_executed)","explanation":"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)."},"skills":["Zero Inputs Have an Explicit Gcd Convention without Division by Zero"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}