# GCD of Integer and Divisor

## Theorem

Let $a, b \in \Z_{>0}$ be strictly positive integers.

Then:

$a \divides b \implies \gcd \set {a, b} = a$

## Proof

We have:

$a \divides b$ by hypothesis
$a \divides a$ from Integer Divides Itself.

Thus $a$ is a common divisor of $a$ and $b$.

$\forall x \in \Z: x \divides a \implies x \le \size a$

As $a$ and $b$ are both positive, the result follows.

$\blacksquare$