Product of GCD and LCM/Proof 2

Proof
Let $a = g m$ and $b = g n$, where $g = \gcd \set {a, b}$ and $m$ and $n$ are coprime.

The existence of $m$ and $n$ are proved by Integers Divided by GCD are Coprime.

It follows that:
 * $a = g m \divides g m n$

and:
 * $b = g n \divides g m n$

So $g m n$ is a common multiple of $a$ and $b$.

Hence there exists an integer $g k \le g m n$ that is divisible by both $a$ and $b$.

Then:

As $g k \le g m n$, it follows that:
 * $k \le m n$

But $m, n$ are coprime.

So: