Product of Coprime Factors

Theorem
Let $$a, b, c \in \Z$$ such that $$a$$ and $$b$$ are coprime.

Let both $$a$$ and $$b$$ be divisors of $$c$$.

Then $$ab$$ is also a divisor of $$c$$.

That is:
 * $$a \perp b \land a \backslash c \land b \backslash c \Longrightarrow a b \backslash c$$.

Proof
We have:
 * $$a \backslash c \Longrightarrow \exists r \in \mathbb{Z}: c = a r$$;
 * $$b \backslash c \Longrightarrow \exists s \in \mathbb{Z}: c = b s$$.

So:

$$ $$ $$ $$ $$ $$