Show that $abc=[ab,bc,ca]*(a,b,c)=(ab,bc,ca)*[a,b,c]$ Let $a,b \in \mathbb N$. Show that
$$ abc=[ab,bc,ca](a,b,c)=(ab,bc,ca)[a,b,c] .$$
How would I prove this?

let $n_a, n_b, n_c$ denote a prime power of $a,b,c$, so the equation is just $$\max(n_a+n_b, n_b+n_c,n_a+n_c)+\min(n_a,n_b,n_c) = \min(n_a+n_b,n_a+n_c,n_b+n_c)+max(n_a,n_b,n_c)=n_a+n_b+n_c$$
because when $n_a+n_b$ obtains maximum, $n_c$ obtains minimum, and vice versa.
