Bézout's Lemma/Proof 5

Proof
Let $\gcd \set {a, b} = d$.

Let $\dfrac a d = p$ and $\dfrac b d = q$.

From Integers Divided by GCD are Coprime:
 * $\gcd \left\{{p, q}\right\} = 1$

From Integer Combination of Coprime Integers:
 * $\exists x, y \in \Z: p x + q y = 1$

The result follows by multiplying both sides by $d$.