Product of GCD and LCM/Proof 2

Theorem

 * $\operatorname{lcm} \left\{{a, b}\right\} \times \gcd \left\{{a, b}\right\} = \left|{a b}\right|$

where:
 * $\operatorname{lcm} \left\{{a, b}\right\}$ is the lowest common multiple of $a$ and $b$
 * $\gcd \left\{{a, b}\right\}$ is the greatest common divisor of $a$ and $b$.

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

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

Since $a = g m \mathop \backslash g m n$ and $b = g n \mathop \backslash g m n$, $g m n$ is the LCD of $a$ and $b$.

Then it follows that $\operatorname{lcm} \left\{{a, b}\right\} \times \gcd \left\{{a, b}\right\} = g m n \times g = g m \times g n = \left|{a b}\right|$.