Bézout's Lemma

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

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

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

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

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

Also known as
This result is also known as Bézout's identity.

Some sources omit the accent off the name: Bezout's lemma or Bezout's identity, which may be a mistake.

Also see

 * Integer Combination of Coprime Integers

Applications
Bézout's Lemma is primarily used when finding solutions to linear Diophantine equations, but is also used to find solutions via Euclidean Division Algorithm.

This result can also be applied to the Extended Euclidean Division Algorithm.