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BEzout

Let $a,b,c\in\mathbb Z$ where $d=\gcd(a,b)$ and $c$ is a multiple of $d$. Suppose that $(x=x_0, y=y_0)$ is one particular integer solution to $$ax+by=c.$$ Then the complete set of integer solutions is $$S=\left\{ \left(x=x_0+k\cdot \frac bd, y=y_0-k\cdot \frac ad \right) \mid k\in\mathbb Z \right\}$$

I cannot figure out how to prove this Theorem!! Any help would be appreciated.

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Hint: Let $\gcd(a,b)=d$. Then an integer c has the form $ax+by$ for some $x,y \in \mathbb{Z}$ if and only if c is a multiple of d. Once you prove this, Bezout's Theorem follows.

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