Euclid's Lemma for Prime Divisors

Lemma
Let $p$ be a prime number.

Let $a$ and $b$ be integers such that:
 * $p \divides a b$

where $\divides$ means is a divisor of.

Then $p \divides a$ or $p \divides b$.

Also presented as
Some sources present this as:

Let $p$ be a prime number.

Let $a$ and $b$ be integers such that:
 * $a b \equiv 0 \pmod p$

Then either $a \equiv 0 \pmod p$ or $b \equiv 0 \pmod p$.

Also known as
Some sources give the name of this as Euclid's first theorem.

Also see
Some sources use this property to define a prime number.