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I am trying to understand Diophantine equation article in wiki. They say that in the given equation:

$$ax + by = c$$

There will be such integers $x,y$ if and only if $c$ is a multiplier of greatest common divisor of $a$ and $b$.

So how does this example lay out with that rule?

$3*3 + 2*4 = 17$

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Here you have $a=3, b=2$. The greatest common divisor of $3$ and $2$ is $1$. $17$ is a multiple of $1$, so there is a solution to the equation, namely that $x=3, y=4$.

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It does in fact verify the statement, because $$\mathrm{gcd}(3,2) = \mathrm{gcd}(3,4) = 1$$

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