Bézout's Identity/Proof 1

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

Let $\gcd \left\{{a, b}\right\}$ be the greatest common divisor of $a$ and $b$.

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

That is, $\gcd \left\{{a, b}\right\}$ is an integer combination (or linear combination) of $a$ and $b$.

Furthermore, $\gcd \left\{{a, b}\right\}$ is the smallest positive integer combination of $a$ and $b$.

Proof
Work the Euclidean Division Algorithm backwards.

Source

 * : $\S 12.3$