# GCD with Prime

## Theorem

Let $p$ be a prime number.

Then:

$\forall n \in \Z: \gcd \set {n, p} = \begin{cases} p & : p \divides n \\ 1 & : p \nmid n \end{cases}$

## Proof

The only divisors of $p$ are $1$ and $p$ itself by definition.

$\gcd \set {n, p} = p$ if and only if $p$ divides $n$.

Hence the result.

$\blacksquare$