Divisors of Product of Coprime Integers

Theorem
Let $a, b, c \in \Z$ be integers.

Let the symbol $\divides$ denote the divisibility relation.

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

Then $a = r s$, where $r \divides b$ and $s \divides c$.

Proof
Let $r = \gcd \set {a, b}$.

By Integers Divided by GCD are Coprime:
 * $\exists s, t \in \Z: a = r s \land b = r t \land \gcd \set {s, t} = 1$

So we have written $a = r s$ where $r$ divides $b$.

We now show that $s$ divides $c$.

Since $a$ divides $b c$ there exists $k$ such that $b c = k a$.

Substituting for $a$ and $b$:
 * $r t c = k r s$

which gives:
 * $t c = k s$

So $s$ divides $t c$.

But we have that:
 * $s \perp t$.

Hence by Euclid's Lemma $s \divides c$ as required.