GCD with Prime

Theorem

Let $p$ be a prime number.

Then:

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


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.

$\blacksquare$

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