Bézout's Identity/Proof 3

Proof
First we establish that:


 * $\exists x, y \in \Z: a x + b y = \gcd \left\{ {a, b}\right\}$

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:


 * $\forall x \in \Z: \exists m, n \in \Z: x = m x + n y \ne 0 \implies d \mathrel \backslash x$

include George E. Andrews: Number Theory coll.2-2