Signed Bezout Witnesses, Complete Integer Solutions and Double-Zero Cases
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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.
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Linear Diophantine equation · qf-mod37-dioph-degenerate-boundary · candidate
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