Product of GCD and LCM

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
It is sufficient to prove that $$\operatorname{lcm} \left\{{a, b}\right\} \times \gcd \left\{{a, b}\right\} = a b$$, where $$a, b \in \Z^*_+$$.

$$ $$

$$ $$ $$ $$ $$

Now we have $$a \backslash m \and b \backslash m \implies m = a r = b s$$.

Also, $$d = a x + b y$$.

So:

$$ $$ $$ $$

So $$m = n \left({s x + r y}\right)$$.

Thus $$n \backslash m \implies n \le \left|{m}\right|$$, while $$a b = d n = \gcd \left\{{a, b}\right\} \times \operatorname{lcm} \left\{{a, b}\right\}$$ as required.

Looks a bit messy to me.