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 $a=gm$ and $b=gn$, 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 $a = gm \backslash gmn$ and $b = gn \backslash gmn$, $gmn$ is the LCD of $a$ and $b$.

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