For all non-zero positive integers $a, b, c$ prove $[a, b, c] = \frac{abc (a, b, c)}{(a, b) (b, c) (c, a)}$

Question

they could help me, I have applied the definition of mcd and mcm but there is a construction that I do not achieve

Answer 1

Hint: Let $p$ be any prime and without loss of generality order $a,b,c$ so that $\nu_p(a)\geq \nu_p(b)\geq \nu_p(c)$ ($\nu_p(n)$ is the number of times $p$ divides $n$ for any positive integer $n$ and prime $p$). Can you show that the number of times $p$ divides the left side is the same as the number of times $p$ divides the right side?