State/Linear Diophantine equations can be solved with the Euclidean algorithm

 =Proof=

Equations in two variables
We begin by proving that if $a,b \in \ZZ$, and $a \neq 0$, $g = GCD(a,b)$, and $m \in \ZZ$, then there exist $x,y \in \ZZ$ such that $ax + by = m$ if and only if $m$ is a multiple of $g$.

Equations in many variables
Now, we prove the solubility criterion for linear Diophantine equations of many variables, using induction on the number of variables.

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