Divisors of Product of Coprime Integers/Corollary

Corollary to Divisors of Product of Coprime Integers

Let $p$ be a prime.

Let $p \divides b c$, where $b \perp c$.

Then $p \divides b$ or $p \divides c$, but not both.

Proof

From the main result, $p = r s$, where $r \divides b$ and $s \divides c$.

But as $p$ is prime, either:

$r = 1$ and $s = p$, or:

$r = p$ and $s = 1$.

So $p \divides b$ or $p \divides c$.

But $p$ can not divide both, as $b \perp c$.

$\blacksquare$