we are given that $a = p_1^{\alpha 1} .... p_k^{\alpha k}$ and $b = p_1^{\beta 1} .... p_k^{\beta k}$. Where $p_1 ... p_k$ are pairwise distinct primes and $\alpha_i$ and $\beta_i$ are non negative integers.
I tried expanding out on the left hand side using the formula for the LCM but i couldn't see where to go from there.
Let $v_p(x)$ be the exponent of a prime $p$ in the factorization of $x$ ($v_p(x)=0$ is $p$ does not divide $x$).
The key observation is that $x$ divides $y$ iff $v_p(x)\le v_p(y)$ for all primes $p$.
From this, it follows that $m$ is a common multiple of $a$ and $b$ iff $v_p(m)\ge v_p(a)$ and $v_p(m)\ge v_p(b)$, or equivalently, $v_p(m)\ge \max(v_p(a), v_p(b))$.
It also follows the dual condition $v_p(\gcd(a,b))=\min(v_p(a), v_p(b))$.
