Bézout's Identity

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 $b$.

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

Applications
It is primarily used when 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.

However, there are sources which suggest that this identity was first noticed by Claude Gaspard Bachet de Méziriac.

Bézout's contribution was to prove a more general result, for polynomials.

Source

 * : $\S 2.2$: Corollary $2.1$