Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$ I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much 

Easy proof by set theory: (I'm a beginner and thought of this, please correct if me if I'm wrong)
Let $a=xy, b=xz$.
So
$a = \{x,y\}$ and $b = \{x, y\}$,
where $x$ is the product of the common prime factors of $a, b$ which is $\gcd(a,b)$ and the product $xyz$ produces $\text{lcm}(a,b)$
By Set Theory,
$P(a \cup b) \cdot P(a \cap b) = P(a) \cdot P(b)$.
$xyz \cdot x = xy \cdot xz$
lcm(a,b) * gcd(a,b) = ab
