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  "body": "\n## Summary\n\nA linear Diophantine equation in two unknowns is an equation $a x + b y = c$ with given integers $a$, $b$ and $c$.\nIts solutions are the pairs of integers $(x, y)$ that satisfy it.\nWhen $a$ and $b$ are not both $0$, the equation has an integer solution if and only if $\\gcd(a, b)$ divides $c$.\nIn this case, let $d = \\gcd(a, b)$ and let $(x_0, y_0)$ be one solution. Then the solutions are the pairs $(x_0 + k b / d, \\; y_0 - k a / d)$ with $k$ an integer.\n\n## Remarks\n\n[[Bézout's identity|0DQVHMYAGA]] gives the existence of a solution, and the [[extended Euclidean algorithm|0964YQFSA8]] computes one.\nFor example, $6 x + 10 y = 7$ has no integer solution, because $\\gcd(6, 10) = 2$ does not divide $7$.\nThe equation is a special case of a [[Diophantine equation|08B1RTW7VZ]].\n\n## Explanations\n\n### L1 human\n\nAsk whether you can pay $c$ with coins of value $a$ and $b$, when you can also get coins back as change.\nEach amount that you can pay is a multiple of $\\gcd(a, b)$, and each such multiple is possible.\nFor example, $3 x + 5 y = 1$ has the solution $x = 2$, $y = -1$.\n",
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