Bézout's Identity

The following is a basic form of Bézout's Identity.

Theorem
Suppose: $$a,b,d \in \Z$$

If $$\gcd(a,b) = d$$, then $$\exists x, y \in \Z$$ such that $$ax + by = d$$.

Proof
Work the Euclidean Division Algorithm backwards.

Applications
It is primarily used with finding solutions to linear Diophantine equations, but is also used to find solutions via Euclidean Division Algorithm.

This Identity/Lemma can also be applied to the Extended Euclidean Division Algorithm.