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.

Since $a = g m \divides g m n$ and $b = g n \divides g m n$, $g m n$ is the LCM of $a$ and $b$.

Then it follows that:
 * $\lcm \set {a, b} \times \gcd \set {a, b} = g m n \times g = g m \times g n = \size {a b}$