GCD with Prime

Theorem
Let $$p$$ be a prime number.

Then:
 * $$\forall n \in \Z: \gcd \left\{{n, p}\right\} = \begin{cases}

p & : p \backslash n \\ 1 & : p \nmid n \end{cases} $$

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

$$\gcd \left\{{n, p}\right\} = p$$ precisely when $$p$$ divides $$n$$.

Hence the result.