Bézout's Identity

The following is a basic form of Bezout'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 used to find solutions via Euclidean Division Algorithm. This Identity/Lemma can be applied to apply the Extended Euclidean Division Algorithm