{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r06-linear-diophantine-equation-0001","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91c21b03c117a06a17b3efb8b5e2a426e7a5429d374c9985ad34f42f563e242e","family_id":"qf-mod37-dioph-solvability","seed":202611066001,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":12,\"b\":18,\"c\":12}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=12, b=18, c=12 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(12, 18, 12, 6, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (12, 18, 12, 6, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0002","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ea60cda2b266a6a0acfbfb5d3bb707ac5d4f0276a287d90681197bbbfd18a7","family_id":"qf-mod37-dioph-solvability","seed":202611066002,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-18,\"b\":27,\"c\":19}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-18, b=27, c=19 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-18, 27, 19, 9, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-18, 27, 19, 9, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0003","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19aff75ea48c8b1034c9405696dd7f843129d0545316c8953d841f143fa9c26c","family_id":"qf-mod37-dioph-solvability","seed":202611066003,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":24,\"b\":36,\"c\":24}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=24, b=36, c=24 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(24, 36, 24, 12, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (24, 36, 24, 12, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0004","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b835e7ce5b7e30c7a6d1e7171a7e53255b4a48e301e59f90f869d70a1598492d","family_id":"qf-mod37-dioph-solvability","seed":202611066004,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-30,\"b\":45,\"c\":31}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-30, b=45, c=31 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-30, 45, 31, 15, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-30, 45, 31, 15, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0005","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87d497cc31c1cb4a86a200e5082b41019ad3d09cc8c167732b07266db474ebfd","family_id":"qf-mod37-dioph-solvability","seed":202611066005,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":36,\"b\":54,\"c\":36}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=36, b=54, c=36 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(36, 54, 36, 18, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (36, 54, 36, 18, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0006","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66c94a9a939f0b1fc252df05672145fd95c980dbc92800868a22faf13e1ea8c3","family_id":"qf-mod37-dioph-solvability","seed":202611066006,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-42,\"b\":63,\"c\":43}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-42, b=63, c=43 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-42, 63, 43, 21, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-42, 63, 43, 21, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0007","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b07669bf6a6d214eb17d1e8af3a01368ff5420e9eeaf4147f1920bc87af6cb16","family_id":"qf-mod37-dioph-solvability","seed":202611066007,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":48,\"b\":72,\"c\":48}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=48, b=72, c=48 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(48, 72, 48, 24, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (48, 72, 48, 24, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0008","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f671c86cb7faf7720a2d1e2069ac29ce7c5499614af78c11ea90fec96c799c88","family_id":"qf-mod37-dioph-solvability","seed":202611066008,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-54,\"b\":81,\"c\":55}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-54, b=81, c=55 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-54, 81, 55, 27, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-54, 81, 55, 27, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0009","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85a836297045db5b3314b9065d936d44a238ff85dc01257844f2628d1544e0fd","family_id":"qf-mod37-dioph-solvability","seed":202611066009,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":60,\"b\":90,\"c\":60}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=60, b=90, c=60 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(60, 90, 60, 30, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (60, 90, 60, 30, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0010","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b540640d8299f4349339a729026be4fb17e6f702d6de28046e4aaa5c48309597","family_id":"qf-mod37-dioph-solvability","seed":202611066010,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-66,\"b\":99,\"c\":67}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-66, b=99, c=67 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-66, 99, 67, 33, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-66, 99, 67, 33, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0011","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b48d5d501bfa999b6f824d8a030522fc1bad84bf19ff1e14a9215581a349d170","family_id":"qf-mod37-dioph-solvability","seed":202611066011,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":72,\"b\":108,\"c\":72}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=72, b=108, c=72 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(72, 108, 72, 36, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (72, 108, 72, 36, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0012","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1058ea2f1812e02100c0103e01bf27bdfc8b5348fdeb7da30b0fa8160de11ed6","family_id":"qf-mod37-dioph-solvability","seed":202611066012,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-78,\"b\":117,\"c\":79}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-78, b=117, c=79 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-78, 117, 79, 39, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-78, 117, 79, 39, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0013","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89380800f3c60c37524fff5005679333b6975f64599b459e108bbdfabd7be806","family_id":"qf-mod37-dioph-solvability","seed":202611066013,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":84,\"b\":126,\"c\":84}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=84, b=126, c=84 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(84, 126, 84, 42, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (84, 126, 84, 42, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0014","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"951b8e34564ed9d2d1f3199a4a3f10dea6c480f66a427c0815d9a73d082c7f93","family_id":"qf-mod37-dioph-solvability","seed":202611066014,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-90,\"b\":135,\"c\":91}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-90, b=135, c=91 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-90, 135, 91, 45, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-90, 135, 91, 45, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0015","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63b58ae7dbb40dc8d3dda1e50704b0b399b42acc09ee826828034e38fc4fc9e3","family_id":"qf-mod37-dioph-solvability","seed":202611066015,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":96,\"b\":144,\"c\":96}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=96, b=144, c=96 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(96, 144, 96, 48, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (96, 144, 96, 48, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0016","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b3214b4b1d519e1afb4092b672966e0e29113a5140f88275a96ac4504ef02e7","family_id":"qf-mod37-dioph-solvability","seed":202611066016,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-102,\"b\":153,\"c\":103}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-102, b=153, c=103 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-102, 153, 103, 51, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-102, 153, 103, 51, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0017","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"760e6c9b7e5343e04c609b89a10a96107c6d341e0c33e0fddad8fd41fd815e30","family_id":"qf-mod37-dioph-solvability","seed":202611066017,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":108,\"b\":162,\"c\":108}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=108, b=162, c=108 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(108, 162, 108, 54, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (108, 162, 108, 54, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0018","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9d7893fcd920f51ddd998ae3b52c80ae0ce422b97d1d7992bc4988ced2bbe02","family_id":"qf-mod37-dioph-solvability","seed":202611066018,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-114,\"b\":171,\"c\":115}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-114, b=171, c=115 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-114, 171, 115, 57, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-114, 171, 115, 57, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0019","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d684b3107bc6c75ab7f016ec98187d3b9e93ebaaf1da66545f36b1412b204ea6","family_id":"qf-mod37-dioph-solvability","seed":202611066019,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":120,\"b\":180,\"c\":120}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=120, b=180, c=120 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(120, 180, 120, 60, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (120, 180, 120, 60, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0020","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d7815ea25a85ae16b4a11f16e9ee43e31e7709330e74ab148f7f83ba96672a7","family_id":"qf-mod37-dioph-solvability","seed":202611066020,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-126,\"b\":189,\"c\":127}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-126, b=189, c=127 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-126, 189, 127, 63, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-126, 189, 127, 63, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0021","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50f8ac71f01d5942db487998e3a80fde36c22df64cc8acbaba9c15575dc06f39","family_id":"qf-mod37-dioph-solvability","seed":202611066021,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":132,\"b\":198,\"c\":132}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=132, b=198, c=132 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(132, 198, 132, 66, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (132, 198, 132, 66, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0022","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"019cbe908941718985870b6a2d9809b1dfd86041748e9aac26e28b91e40ea9a4","family_id":"qf-mod37-dioph-solvability","seed":202611066022,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-138,\"b\":207,\"c\":139}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-138, b=207, c=139 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-138, 207, 139, 69, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-138, 207, 139, 69, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0023","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe93a9860679d8072ffa2430a005372e21f741d10a991367d79b07d45910b7f","family_id":"qf-mod37-dioph-solvability","seed":202611066023,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":144,\"b\":216,\"c\":144}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=144, b=216, c=144 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(144, 216, 144, 72, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (144, 216, 144, 72, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0024","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c999d2fc4b387d406a3a3b79b2fbe308f98dafc1c4dac7793db619821faff784","family_id":"qf-mod37-dioph-solvability","seed":202611066024,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":-150,\"b\":225,\"c\":151}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=-150, b=225, c=151 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(-150, 225, 151, 75, No, No, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (-150, 225, 151, 75, No, No, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0025","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d971bb9207b3e23b224d64bbf2f1b44eb43be507a010202926bbf5d2290d9ecd","family_id":"qf-mod37-dioph-solvability","seed":202611066025,"semantic_template":"qf-mod37-dioph-solvability/{\"a\":156,\"b\":234,\"c\":156}"},"fields":{"title":"Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion","stem":"Use exact original integers a=156, b=234, c=156 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (a, b, c, gcd, both_coefficients_zero, solvable, boundary).","answer":{"kind":"expression","value":"(156, 234, 156, 78, No, Yes, gcd_divides_c_iff_integer_solution)","explanation":"Return original coefficients, normalized gcd, the double-zero flag and solvability. With both coefficients zero, use c=0 rather than any divisibility by zero test. The exact result, retaining the requested order, is (156, 234, 156, 78, No, Yes, gcd_divides_c_iff_integer_solution)."},"skills":["Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion"],"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-20261106-r06-linear-diophantine-equation-0026","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeb2e0681454b5623e6e2ecbe56bb280794f2a2231e12d484712e36ecef211f4","family_id":"qf-mod37-dioph-particular-solution","seed":202611066026,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":12,\"b\":18,\"c\":6}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=12, b=18, c=6 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(6, -1, 1, -1, 1, 6, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (6, -1, 1, -1, 1, 6, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0027","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6363c3153bb99c9ca9495681401f2c5cf92de88168775313b2917e347759c9a","family_id":"qf-mod37-dioph-particular-solution","seed":202611066027,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-18,\"b\":27,\"c\":18}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-18, b=27, c=18 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(9, 1, 1, 2, 2, 18, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (9, 1, 1, 2, 2, 18, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0028","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ac450171a7e87ac0954a3ca51c47de3bc7660fa9a21e03ce96b1205319964d9","family_id":"qf-mod37-dioph-particular-solution","seed":202611066028,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":24,\"b\":36,\"c\":36}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=24, b=36, c=36 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(12, -1, 1, -3, 3, 36, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (12, -1, 1, -3, 3, 36, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0029","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ef61cb03a9a87d3e57825133899ada2a6e6a30dc2280ad21296589626839c9f","family_id":"qf-mod37-dioph-particular-solution","seed":202611066029,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-30,\"b\":45,\"c\":60}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-30, b=45, c=60 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(15, 1, 1, 4, 4, 60, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (15, 1, 1, 4, 4, 60, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0030","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0e2e6a40c82aa61a349620914576efea30c8cba3c2831ee739551d7d19d87c7","family_id":"qf-mod37-dioph-particular-solution","seed":202611066030,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":36,\"b\":54,\"c\":90}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=36, b=54, c=90 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(18, -1, 1, -5, 5, 90, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (18, -1, 1, -5, 5, 90, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0031","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06c88da9a8e2fded26b45ef87d26b72f1eb8c46533529338a3d0d347c79b57f3","family_id":"qf-mod37-dioph-particular-solution","seed":202611066031,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-42,\"b\":63,\"c\":126}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-42, b=63, c=126 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(21, 1, 1, 6, 6, 126, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (21, 1, 1, 6, 6, 126, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0032","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86a824b4293f67bfde8aff6bea0089339f0a585fd98eb7963c9196a75608da5f","family_id":"qf-mod37-dioph-particular-solution","seed":202611066032,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":48,\"b\":72,\"c\":168}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=48, b=72, c=168 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(24, -1, 1, -7, 7, 168, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (24, -1, 1, -7, 7, 168, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0033","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38337b84e43c17f6dc514c111e6a5028640d5ba364f8ffc364600c6c3e177fe0","family_id":"qf-mod37-dioph-particular-solution","seed":202611066033,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-54,\"b\":81,\"c\":216}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-54, b=81, c=216 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(27, 1, 1, 8, 8, 216, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (27, 1, 1, 8, 8, 216, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0034","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6148089268e2346fbb41b8b779308f6be0c38cd386bd401fdb4d4768978b8dc","family_id":"qf-mod37-dioph-particular-solution","seed":202611066034,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":60,\"b\":90,\"c\":270}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=60, b=90, c=270 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(30, -1, 1, -9, 9, 270, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (30, -1, 1, -9, 9, 270, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0035","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d2dc24fe533dc34aa965be39823e6963dcf6e07afda98061190d42cab53075e","family_id":"qf-mod37-dioph-particular-solution","seed":202611066035,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-66,\"b\":99,\"c\":330}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-66, b=99, c=330 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(33, 1, 1, 10, 10, 330, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (33, 1, 1, 10, 10, 330, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0036","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37807e3696b11fc322981f83b9ac7294500d00508347ae7363223d04c71aca6c","family_id":"qf-mod37-dioph-particular-solution","seed":202611066036,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":72,\"b\":108,\"c\":396}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=72, b=108, c=396 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(36, -1, 1, -11, 11, 396, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (36, -1, 1, -11, 11, 396, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0037","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d3d87a33e83f0315d2427728dac8398db8d2014ad563b6e96fe525d49d5ba2f","family_id":"qf-mod37-dioph-particular-solution","seed":202611066037,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-78,\"b\":117,\"c\":468}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-78, b=117, c=468 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(39, 1, 1, 12, 12, 468, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (39, 1, 1, 12, 12, 468, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0038","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0afbedfc67e5c9cf459878d67d94187259a3152a0228e4760c49096302693a57","family_id":"qf-mod37-dioph-particular-solution","seed":202611066038,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":84,\"b\":126,\"c\":546}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=84, b=126, c=546 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(42, -1, 1, -13, 13, 546, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (42, -1, 1, -13, 13, 546, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0039","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d12e10bb9ac51c3987d1032de9dbc57c909d5af84d4573d221c6f6905b6f7f17","family_id":"qf-mod37-dioph-particular-solution","seed":202611066039,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-90,\"b\":135,\"c\":630}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-90, b=135, c=630 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(45, 1, 1, 14, 14, 630, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (45, 1, 1, 14, 14, 630, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0040","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b15521fa27c60581966a8b11e4b78e637b6f1a14bc48a9217ef42a845ba66cb8","family_id":"qf-mod37-dioph-particular-solution","seed":202611066040,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":96,\"b\":144,\"c\":720}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=96, b=144, c=720 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(48, -1, 1, -15, 15, 720, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (48, -1, 1, -15, 15, 720, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0041","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68e3aa7238bf7837be1e9234e81a14c6b39474452d1e580fce5c287666dab42a","family_id":"qf-mod37-dioph-particular-solution","seed":202611066041,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-102,\"b\":153,\"c\":816}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-102, b=153, c=816 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(51, 1, 1, 16, 16, 816, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (51, 1, 1, 16, 16, 816, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0042","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4995e424f8c397538eb6c617eebcb544e3a35284365a7ff7991a9bc3e341d23","family_id":"qf-mod37-dioph-particular-solution","seed":202611066042,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":108,\"b\":162,\"c\":918}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=108, b=162, c=918 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(54, -1, 1, -17, 17, 918, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (54, -1, 1, -17, 17, 918, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0043","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fc1cb2a259f783fb57bbeabaf4f2cb07dd6261e8a610cc8a99672d4b15cdd3c","family_id":"qf-mod37-dioph-particular-solution","seed":202611066043,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-114,\"b\":171,\"c\":1026}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-114, b=171, c=1026 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(57, 1, 1, 18, 18, 1026, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (57, 1, 1, 18, 18, 1026, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0044","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0580ddef60142d2c11988d77ecefc40ca18e09ed9e57fb838184a49d163f06e","family_id":"qf-mod37-dioph-particular-solution","seed":202611066044,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":120,\"b\":180,\"c\":1140}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=120, b=180, c=1140 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(60, -1, 1, -19, 19, 1140, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (60, -1, 1, -19, 19, 1140, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0045","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b64678e8ee9fa564b9cee04d255804b27f01d5270e551cf3959d7b90187c9b","family_id":"qf-mod37-dioph-particular-solution","seed":202611066045,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-126,\"b\":189,\"c\":1260}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-126, b=189, c=1260 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(63, 1, 1, 20, 20, 1260, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (63, 1, 1, 20, 20, 1260, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0046","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21d11bee3add551e69727c4da6fd192f0ed913ec92bf2a64a69169a22b9a4670","family_id":"qf-mod37-dioph-particular-solution","seed":202611066046,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":132,\"b\":198,\"c\":1386}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=132, b=198, c=1386 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(66, -1, 1, -21, 21, 1386, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (66, -1, 1, -21, 21, 1386, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0047","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92dcb826012d4ec8d70918d3cb50c1c73b967a86588fdb5239248d987198a949","family_id":"qf-mod37-dioph-particular-solution","seed":202611066047,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-138,\"b\":207,\"c\":1518}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-138, b=207, c=1518 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(69, 1, 1, 22, 22, 1518, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (69, 1, 1, 22, 22, 1518, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0048","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"013e1162a1fc200f31db00b3321713a91ffb5e769582b0e0ea6c3787d90be115","family_id":"qf-mod37-dioph-particular-solution","seed":202611066048,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":144,\"b\":216,\"c\":1656}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=144, b=216, c=1656 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(72, -1, 1, -23, 23, 1656, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (72, -1, 1, -23, 23, 1656, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0049","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b86d8a80f11703be0f90c0c90bba2b45d485ea45f5cebcc14f6e342ac1cebd51","family_id":"qf-mod37-dioph-particular-solution","seed":202611066049,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":-150,\"b\":225,\"c\":1800}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=-150, b=225, c=1800 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(75, 1, 1, 24, 24, 1800, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (75, 1, 1, 24, 24, 1800, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0050","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4079748974e46aa6827ce6d41ee7e169e32c904d04bfecb1e7ca5130f6eac619","family_id":"qf-mod37-dioph-particular-solution","seed":202611066050,"semantic_template":"qf-mod37-dioph-particular-solution/{\"a\":156,\"b\":234,\"c\":1950}"},"fields":{"title":"Scale a Signed Bezout Pair to Obtain One Integer Solution","stem":"Use exact original integers a=156, b=234, c=1950 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate).","answer":{"kind":"expression","value":"(78, -1, 1, -25, 25, 1950, Yes, extended_Euclidean_Bezout_certificate)","explanation":"Require a nondegenerate solvable equation. Give Bezout coefficients for d, scale them by c/d to obtain x0,y0 and check a*x0+b*y0=c. This constructs one solution, not the entire solution set. The exact result, retaining the requested order, is (78, -1, 1, -25, 25, 1950, Yes, extended_Euclidean_Bezout_certificate)."},"skills":["Scale a Signed Bezout Pair to Obtain One Integer Solution"],"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-20261106-r06-linear-diophantine-equation-0051","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"972154bae865444cf39eba9217801bae9ee4afc4586d603634e06fd159dedb1e","family_id":"qf-mod37-dioph-general-solution","seed":202611066051,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":12,\"b\":18,\"c\":6}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=12, b=18, c=6 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-1, 1, 3, -2, x=-1+(3)t, y=1+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-1, 1, 3, -2, x=-1+(3)t, y=1+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0052","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e5041ecab803a1eb38a8faebf30747245fe12114e80a4e64020123a3d512e0","family_id":"qf-mod37-dioph-general-solution","seed":202611066052,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-18,\"b\":27,\"c\":18}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-18, b=27, c=18 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(2, 2, 3, 2, x=2+(3)t, y=2+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (2, 2, 3, 2, x=2+(3)t, y=2+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0053","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d61ee19c531746d7e520ceeb4449e41d1c168a4d6e0d2290d4907d8bdb432f15","family_id":"qf-mod37-dioph-general-solution","seed":202611066053,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":24,\"b\":36,\"c\":36}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=24, b=36, c=36 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-3, 3, 3, -2, x=-3+(3)t, y=3+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-3, 3, 3, -2, x=-3+(3)t, y=3+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0054","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a44d258da36e943394b4de9d5fbb697a5a613fec7da9a2b5678c0fa5900be5","family_id":"qf-mod37-dioph-general-solution","seed":202611066054,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-30,\"b\":45,\"c\":60}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-30, b=45, c=60 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(4, 4, 3, 2, x=4+(3)t, y=4+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (4, 4, 3, 2, x=4+(3)t, y=4+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0055","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16f8fcec8e79b0fd99b5421f9ffa20077070321b3a22a051fd0b6814e70a1097","family_id":"qf-mod37-dioph-general-solution","seed":202611066055,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":36,\"b\":54,\"c\":90}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=36, b=54, c=90 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-5, 5, 3, -2, x=-5+(3)t, y=5+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-5, 5, 3, -2, x=-5+(3)t, y=5+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0056","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"659e991d1a60f8c325f456bb4bb6c77c3fe8cf0745456e588ecf604ac093745a","family_id":"qf-mod37-dioph-general-solution","seed":202611066056,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-42,\"b\":63,\"c\":126}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-42, b=63, c=126 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(6, 6, 3, 2, x=6+(3)t, y=6+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (6, 6, 3, 2, x=6+(3)t, y=6+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0057","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9b5e4e6da306d07cbb7b54edb78d0dc8e62838e0ddf287f24f803772353a157","family_id":"qf-mod37-dioph-general-solution","seed":202611066057,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":48,\"b\":72,\"c\":168}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=48, b=72, c=168 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-7, 7, 3, -2, x=-7+(3)t, y=7+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-7, 7, 3, -2, x=-7+(3)t, y=7+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0058","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea89d253f1397921b3c94f9f383aea3789a402a808a50dd9734066433ba02e7b","family_id":"qf-mod37-dioph-general-solution","seed":202611066058,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-54,\"b\":81,\"c\":216}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-54, b=81, c=216 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(8, 8, 3, 2, x=8+(3)t, y=8+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (8, 8, 3, 2, x=8+(3)t, y=8+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0059","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78f69b11e4df17e10f8016c6950aac95f424406f43b1da9d600a67c821455417","family_id":"qf-mod37-dioph-general-solution","seed":202611066059,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":60,\"b\":90,\"c\":270}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=60, b=90, c=270 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-9, 9, 3, -2, x=-9+(3)t, y=9+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-9, 9, 3, -2, x=-9+(3)t, y=9+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0060","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1adb472fa9b39752f25eeb3eb638132dda71859461299aad8c50f93a273e782","family_id":"qf-mod37-dioph-general-solution","seed":202611066060,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-66,\"b\":99,\"c\":330}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-66, b=99, c=330 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(10, 10, 3, 2, x=10+(3)t, y=10+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (10, 10, 3, 2, x=10+(3)t, y=10+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0061","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72848c46266d8f0e1df59ee7ad604284ef6cf7adc534b3e993d3b568fe3b0b5e","family_id":"qf-mod37-dioph-general-solution","seed":202611066061,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":72,\"b\":108,\"c\":396}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=72, b=108, c=396 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-11, 11, 3, -2, x=-11+(3)t, y=11+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-11, 11, 3, -2, x=-11+(3)t, y=11+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0062","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a10f28a3cfa7e8bcc2e4ee968e4e29c29a839ac8382a4b80c30658e9f8757a1a","family_id":"qf-mod37-dioph-general-solution","seed":202611066062,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-78,\"b\":117,\"c\":468}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-78, b=117, c=468 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(12, 12, 3, 2, x=12+(3)t, y=12+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (12, 12, 3, 2, x=12+(3)t, y=12+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0063","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca59694950b19b27e956fd50358c139f0351ddf4f876bd8369bcefa51c50276f","family_id":"qf-mod37-dioph-general-solution","seed":202611066063,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":84,\"b\":126,\"c\":546}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=84, b=126, c=546 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-13, 13, 3, -2, x=-13+(3)t, y=13+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-13, 13, 3, -2, x=-13+(3)t, y=13+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0064","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbca5033bd88cf4483e307e6762415892009d6fead99a75cba8b6bccf0a1da0e","family_id":"qf-mod37-dioph-general-solution","seed":202611066064,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-90,\"b\":135,\"c\":630}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-90, b=135, c=630 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0065","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"356524e1771c07ef61afd3a69d89441b4f18084306325c3be2fc79b827827211","family_id":"qf-mod37-dioph-general-solution","seed":202611066065,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":96,\"b\":144,\"c\":720}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=96, b=144, c=720 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-15, 15, 3, -2, x=-15+(3)t, y=15+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-15, 15, 3, -2, x=-15+(3)t, y=15+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0066","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d34a825da33373ef178aa519b5158d8e976be8912c0f10e807d7cc3cbc360460","family_id":"qf-mod37-dioph-general-solution","seed":202611066066,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-102,\"b\":153,\"c\":816}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-102, b=153, c=816 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(16, 16, 3, 2, x=16+(3)t, y=16+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (16, 16, 3, 2, x=16+(3)t, y=16+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0067","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090747e638733a0f60fd9a83335a1c62e1841f3d6144823265d6f308498dd883","family_id":"qf-mod37-dioph-general-solution","seed":202611066067,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":108,\"b\":162,\"c\":918}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=108, b=162, c=918 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-17, 17, 3, -2, x=-17+(3)t, y=17+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-17, 17, 3, -2, x=-17+(3)t, y=17+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0068","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a06340d0b0951ef7fe97c7843933ae95b42ae3cf3daaa00326eb62f7e8bd4185","family_id":"qf-mod37-dioph-general-solution","seed":202611066068,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-114,\"b\":171,\"c\":1026}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-114, b=171, c=1026 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(18, 18, 3, 2, x=18+(3)t, y=18+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (18, 18, 3, 2, x=18+(3)t, y=18+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0069","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30d2b891cd1c24d90374b7177a1ee54b9876be7929d98aaf2b4b1d9077b2ffcf","family_id":"qf-mod37-dioph-general-solution","seed":202611066069,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":120,\"b\":180,\"c\":1140}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=120, b=180, c=1140 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-19, 19, 3, -2, x=-19+(3)t, y=19+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-19, 19, 3, -2, x=-19+(3)t, y=19+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0070","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"093c4f2ad303f0aea3f728210758888f67f131881cef20dc40c6b9eeef312f69","family_id":"qf-mod37-dioph-general-solution","seed":202611066070,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-126,\"b\":189,\"c\":1260}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-126, b=189, c=1260 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(20, 20, 3, 2, x=20+(3)t, y=20+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (20, 20, 3, 2, x=20+(3)t, y=20+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0071","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4c20e77f8c8c51da5505d3d5e29eb334a2dc2c300dea77fb942d759f31a6e2","family_id":"qf-mod37-dioph-general-solution","seed":202611066071,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":132,\"b\":198,\"c\":1386}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=132, b=198, c=1386 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-21, 21, 3, -2, x=-21+(3)t, y=21+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-21, 21, 3, -2, x=-21+(3)t, y=21+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0072","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f14a68aff26e577d1641e62197309ad6a1a7097de44c403928a996930f6ea15","family_id":"qf-mod37-dioph-general-solution","seed":202611066072,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-138,\"b\":207,\"c\":1518}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-138, b=207, c=1518 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(22, 22, 3, 2, x=22+(3)t, y=22+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (22, 22, 3, 2, x=22+(3)t, y=22+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0073","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9850e3edeb5e869758d00c00d21a9814f0797a0b85b3674ad5f45863ff1629f3","family_id":"qf-mod37-dioph-general-solution","seed":202611066073,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":144,\"b\":216,\"c\":1656}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=144, b=216, c=1656 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-23, 23, 3, -2, x=-23+(3)t, y=23+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-23, 23, 3, -2, x=-23+(3)t, y=23+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0074","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cade2bce2afaa2bd2d6746d1265c3afaee6789ad6f1ac54e41c6331e337dc5c9","family_id":"qf-mod37-dioph-general-solution","seed":202611066074,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":-150,\"b\":225,\"c\":1800}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=-150, b=225, c=1800 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(24, 24, 3, 2, x=24+(3)t, y=24+(2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (24, 24, 3, 2, x=24+(3)t, y=24+(2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0075","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47e66380f963e109a62f112e219c30a0e3f819cabdbdea303a77cac1a701f0fe","family_id":"qf-mod37-dioph-general-solution","seed":202611066075,"semantic_template":"qf-mod37-dioph-general-solution/{\"a\":156,\"b\":234,\"c\":1950}"},"fields":{"title":"All Solutions Need the Gcd-Normalized Integer Parameter","stem":"Use exact original integers a=156, b=234, c=1950 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (x0, y0, step_x, step_y, x_formula, y_formula, certificate).","answer":{"kind":"expression","value":"(-25, 25, 3, -2, x=-25+(3)t, y=25+(-2)t, all_and_only_integer_solutions_for_t_in_Z)","explanation":"For a nondegenerate solvable equation use x=x0+(b/d)t and y=y0−(a/d)t for every t∈Z. Retain the exact source signed step values and literal formula strings, even with parenthesized negative or zero steps. This is a whole-integer parameterization, not a finite list of samples. The exact result, retaining the requested order, is (-25, 25, 3, -2, x=-25+(3)t, y=25+(-2)t, all_and_only_integer_solutions_for_t_in_Z)."},"skills":["All Solutions Need the Gcd-Normalized Integer Parameter"],"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-20261106-r06-linear-diophantine-equation-0076","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88fde46c707342ae16d6f804441330ba7055c5995c974598508452788f6725fe","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066076,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-12}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-12 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-12, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-12, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0077","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9406de55e575a75de093d554c2463647297c375228f68c0baed709e685a6990","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066077,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-11}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-11 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-11, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-11, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0078","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5313d9cc51045516b11a2e0bbe750aeef657ad1d86b25271c1380dde1b3a98e1","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066078,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-10}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-10 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-10, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-10, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0079","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"756f6565ac0a6cf0e82abb288c93b75ad88bf268de8bee8a9f7f8747d5f21dfe","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066079,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-9}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-9 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-9, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-9, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0080","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e111a2a6129b4a59ad2fc805830798ab3fdc8b6d5b52cfb1e17445e00ac97b0","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066080,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-8}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-8 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-8, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-8, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0081","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e54301e3696251777bdb29ed64e1bbb88ac37030f29097181cd4fcfee86630a5","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066081,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-7}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-7 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-7, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-7, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0082","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52dc6b89d1c1f76f8c4e31ac33eb3d3f0ff304d49eb9d904d6fc775d508795d3","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066082,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-6}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-6 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-6, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-6, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0083","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b37c734513470909d3b4cffd48c0ba6e0d65b3261ddeab9402796f433682b16","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066083,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-5}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-5 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-5, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-5, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0084","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d080acfb3186b8635b8a220d1906c28372e5ff6f3a3fcabf75fed9c6b94460f2","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066084,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-4}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-4 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-4, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-4, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0085","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03c1b8470749d28cce5e0d3a713a755a3639d376c9ef72986d6d355a3bddb944","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066085,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-3}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-3 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-3, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-3, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0086","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76bfb2151c18fcad318ecebc3eb7a3f09ed9243423df37181b6c67882093357c","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066086,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-2}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-2 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-2, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-2, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0087","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99dd7c3276cc310d63c9832cbb0cfe29f549b82ca202d6ea2864b67ac08a9f7c","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066087,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":-1}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=-1 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(-1, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (-1, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0088","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7654de5620cac9aa67bbef75efa6bd8823c726a5fdf78b3df8836a3011bc0dc","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066088,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":0}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=0 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(0, Yes, all_integer_pairs, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (0, Yes, all_integer_pairs, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0089","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4850d4e3231e86107141e647383dc57e4daaf0eef204745ceaad942c8510a55","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066089,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":1}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=1 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(1, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (1, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0090","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7471f2178c2d23997fdcc3bb52c1a6e95842265b6566d2f799c495c8629becae","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066090,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":2}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=2 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(2, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (2, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0091","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d251cbc1887cf457c3527d0b52ebbe55e8ff38ed0b35c998042f35f18aa867da","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066091,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":3}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=3 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(3, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (3, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0092","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eea4cc771a89254279fdcd750938ec5a30d2f93c69fb1839a20d8caf03c3bfc","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066092,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":4}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=4 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(4, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (4, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0093","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c31298915853ebf17842b62a9add3163d2dbf25c4beb35e0ed88170fe25ab8d5","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066093,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":5}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=5 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(5, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (5, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0094","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7524d1d2b7959c115fe4d9d6f42bd0635badbd5fa381f380d17fc50b4ede1907","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066094,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":6}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=6 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(6, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (6, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0095","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33a9bfd46aaa679b1c2802b1fc212d7c987e33faa2a7eb33184fb24f7e2bc580","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066095,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":7}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=7 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(7, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (7, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0096","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"202b2410131548f4a5dc9820d58ef94d31d040c4cc55e1fa8dc361e32d3f21c0","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066096,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":8}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=8 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(8, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (8, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0097","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"433d49597207618423ce86020018d34a8ede10c2bbe3516a8f004158d71ffd52","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066097,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":9}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=9 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(9, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (9, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0098","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"680095d89cc63c8e89639b7e110f93fb6618333b3584e85a51a7c01276a89e3e","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066098,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":10}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=10 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(10, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (10, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0099","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48f13008b6448e0c609788f5c83b7bae98d33491c1492b92d18d3b20f3aa1291","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066099,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":11}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=11 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(11, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (11, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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-20261106-r06-linear-diophantine-equation-0100","batch_id":"batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fd60954c306e502a62e1b912778d05c36e91b8fb963737aebedc3d452126e59","family_id":"qf-mod37-dioph-degenerate-boundary","seed":202611066100,"semantic_template":"qf-mod37-dioph-degenerate-boundary/{\"a\":0,\"b\":0,\"c\":12}"},"fields":{"title":"Double-Zero Coefficients Must Not Divide by Gcd Zero","stem":"Use exact original integers a=0, b=0, c=12 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).","answer":{"kind":"expression","value":"(12, No, no_integer_solution, never_divide_by_gcd_zero)","explanation":"Require a=b=0. If c=0 every integer pair is a solution; otherwise none is. Keep that separate from the one-parameter nondegenerate formula. The exact result, retaining the requested order, is (12, No, no_integer_solution, never_divide_by_gcd_zero)."},"skills":["Double-Zero Coefficients Must Not Divide by Gcd 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."}}