# Signed Bezout Witnesses, Complete Integer Solutions and Double-Zero Cases

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(12, 18, 12, 6, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0001/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 2. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-18, 27, 19, 9, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0002/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 3. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(24, 36, 24, 12, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0003/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 4. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-30, 45, 31, 15, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0004/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 5. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(36, 54, 36, 18, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0005/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 6. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-42, 63, 43, 21, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0006/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 7. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(48, 72, 48, 24, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0007/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 8. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-54, 81, 55, 27, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0008/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 9. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(60, 90, 60, 30, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0009/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 10. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-66, 99, 67, 33, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0010/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 11. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(72, 108, 72, 36, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0011/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 12. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-78, 117, 79, 39, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0012/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 13. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(84, 126, 84, 42, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0013/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 14. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-90, 135, 91, 45, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0014/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 15. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(96, 144, 96, 48, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0015/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 16. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-102, 153, 103, 51, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0016/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 17. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(108, 162, 108, 54, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0017/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 18. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-114, 171, 115, 57, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0018/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 19. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(120, 180, 120, 60, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0019/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 20. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-126, 189, 127, 63, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0020/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 21. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(132, 198, 132, 66, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0021/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 22. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-138, 207, 139, 69, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0022/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 23. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(144, 216, 144, 72, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0023/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 24. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(-150, 225, 151, 75, No, No, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0024/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 25. Gcd Divisibility Is the Nondegenerate Integer-Solvability Criterion

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).

View answer and explanation(156, 234, 156, 78, No, Yes, gcd_divides_c_iff_integer_solution)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0025/)

Linear Diophantine equation · qf-mod37-dioph-solvability · candidate





## 26. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(6, -1, 1, -1, 1, 6, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0026/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 27. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(9, 1, 1, 2, 2, 18, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0027/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 28. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(12, -1, 1, -3, 3, 36, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0028/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 29. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(15, 1, 1, 4, 4, 60, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0029/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 30. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(18, -1, 1, -5, 5, 90, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0030/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 31. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(21, 1, 1, 6, 6, 126, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0031/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 32. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(24, -1, 1, -7, 7, 168, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0032/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 33. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(27, 1, 1, 8, 8, 216, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0033/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 34. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(30, -1, 1, -9, 9, 270, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0034/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 35. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(33, 1, 1, 10, 10, 330, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0035/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 36. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(36, -1, 1, -11, 11, 396, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0036/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 37. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(39, 1, 1, 12, 12, 468, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0037/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 38. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(42, -1, 1, -13, 13, 546, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0038/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 39. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(45, 1, 1, 14, 14, 630, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0039/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 40. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(48, -1, 1, -15, 15, 720, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0040/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 41. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(51, 1, 1, 16, 16, 816, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0041/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 42. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(54, -1, 1, -17, 17, 918, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0042/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 43. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(57, 1, 1, 18, 18, 1026, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0043/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 44. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(60, -1, 1, -19, 19, 1140, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0044/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 45. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(63, 1, 1, 20, 20, 1260, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0045/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 46. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(66, -1, 1, -21, 21, 1386, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0046/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 47. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(69, 1, 1, 22, 22, 1518, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0047/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 48. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(72, -1, 1, -23, 23, 1656, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0048/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 49. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(75, 1, 1, 24, 24, 1800, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0049/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 50. Scale a Signed Bezout Pair to Obtain One Integer Solution

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).

View answer and explanation(78, -1, 1, -25, 25, 1950, Yes, extended_Euclidean_Bezout_certificate)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0050/)

Linear Diophantine equation · qf-mod37-dioph-particular-solution · candidate





## 51. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-1, 1, 3, -2, x=-1+(3)t, y=1+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0051/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 52. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(2, 2, 3, 2, x=2+(3)t, y=2+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0052/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 53. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-3, 3, 3, -2, x=-3+(3)t, y=3+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0053/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 54. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(4, 4, 3, 2, x=4+(3)t, y=4+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0054/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 55. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-5, 5, 3, -2, x=-5+(3)t, y=5+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0055/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 56. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(6, 6, 3, 2, x=6+(3)t, y=6+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0056/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 57. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-7, 7, 3, -2, x=-7+(3)t, y=7+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0057/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 58. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(8, 8, 3, 2, x=8+(3)t, y=8+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0058/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 59. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-9, 9, 3, -2, x=-9+(3)t, y=9+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0059/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 60. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(10, 10, 3, 2, x=10+(3)t, y=10+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0060/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 61. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-11, 11, 3, -2, x=-11+(3)t, y=11+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0061/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 62. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(12, 12, 3, 2, x=12+(3)t, y=12+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0062/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 63. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-13, 13, 3, -2, x=-13+(3)t, y=13+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0063/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 64. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(14, 14, 3, 2, x=14+(3)t, y=14+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0064/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 65. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-15, 15, 3, -2, x=-15+(3)t, y=15+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0065/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 66. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(16, 16, 3, 2, x=16+(3)t, y=16+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0066/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 67. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-17, 17, 3, -2, x=-17+(3)t, y=17+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0067/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 68. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(18, 18, 3, 2, x=18+(3)t, y=18+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0068/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 69. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-19, 19, 3, -2, x=-19+(3)t, y=19+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0069/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 70. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(20, 20, 3, 2, x=20+(3)t, y=20+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0070/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 71. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-21, 21, 3, -2, x=-21+(3)t, y=21+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0071/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 72. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(22, 22, 3, 2, x=22+(3)t, y=22+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0072/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 73. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-23, 23, 3, -2, x=-23+(3)t, y=23+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0073/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 74. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(24, 24, 3, 2, x=24+(3)t, y=24+(2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0074/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 75. All Solutions Need the Gcd-Normalized Integer Parameter

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).

View answer and explanation(-25, 25, 3, -2, x=-25+(3)t, y=25+(-2)t, all_and_only_integer_solutions_for_t_in_Z)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0075/)

Linear Diophantine equation · qf-mod37-dioph-general-solution · candidate





## 76. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-12, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0076/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 77. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-11, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0077/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 78. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-10, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0078/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 79. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-9, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0079/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 80. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-8, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0080/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 81. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-7, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0081/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 82. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-6, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0082/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 83. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-5, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0083/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 84. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-4, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0084/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 85. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-3, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0085/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 86. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-2, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0086/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 87. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(-1, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0087/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 88. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(0, Yes, all_integer_pairs, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0088/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 89. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(1, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0089/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 90. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(2, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0090/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 91. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(3, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0091/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 92. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(4, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0092/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 93. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(5, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0093/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 94. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(6, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0094/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 95. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(7, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0095/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 96. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(8, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0096/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 97. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(9, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0097/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 98. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(10, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0098/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 99. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(11, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0099/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate





## 100. Double-Zero Coefficients Must Not Divide by Gcd Zero

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).

View answer and explanation(12, No, no_integer_solution, never_divide_by_gcd_zero)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0100/)

Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate



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