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  "body": "\n## Statement\n\nLet $m$, $n$ and $c$ be positive integers. Consider the [[linear Diophantine equation|0R9GF3JD0K]] $a m + b n = c$ in the unknowns $a$ and $b$, where $a$ and $b$ may be any integers, negative ones included.\n\nThis equation has an integer solution $(a, b)$ if and only if $c$ is a multiple of the [[greatest common divisor|02WF878BH1]] of $m$ and $n$.\n\n## Remarks\n\nOne direction is easy. The number $\\gcd(m, n)$ divides $a m + b n$ for all integers $a$ and $b$, so it divides $c$ when a solution exists.\n\nThe other direction follows from [[Bézout's identity|0DQVHMYAGA]]. If $c = k \\gcd(m, n)$, multiply a Bézout pair by $k$.\n\nFor example, $6a + 10b = 7$ has no solution, since $\\gcd(6, 10) = 2$ does not divide $7$. But $6a + 10b = 8$ has the solution $a = -2$, $b = 2$.\n",
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  "name": "Solvability of a linear Diophantine equation in two unknowns",
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