g.c.d. and l.c.m. of three numbers If m,n,k are any three positive integers prove that
$$(m,n)(m,k)(n,k)[m,n,k]^2=[m,n][n,k][m,k](m,n,k)^2$$
where $(a,b)$ Is the g.c.d. of $a$ and $b$ and $[a,b]$ is the l.c.m. of $a$ and $b$.
I don't even know where to start.
This question can be found in $2^{nd}$ chapter (arithmetic of integers) of challenges and thrills of pre-college mathematics. A solution given while keeping this fact in mind would be appreciated.

Let the highest exponent of prime $p$ in $m,n,k$ be $$M,N,K$$ respectively 
WLOG $M\ge N\ge K$
The highest exponent of $p$ in left side will be $$N+K+K+2M$$
and that in the right side, $$M+N+M+2K$$
This holds true for any prime that divides $mnk$
