Bézout's Identity/Proof 3

Let $a,b\in\Z$ such that $a$ and $b$ are not both zero.

Let $d=\gcd\left\{a,b\right\}$

First we establish that $\exists x,y\in\Z:ax+by=\gcd\left\{a,b\right\}$

\emph{include George E. Andrews: Number Theory coll.2-1}

Now to show that $=\gcd\left\{a,b\right\}$ is the smallest positive number to satisfy the equation,

we first show that $\forall x\in\Z,\exists m,n\in\Z x=mx+ny\neq0\implies d\backslash x$

\emph{include George E. Andrews: Number Theory coll.2-2}