Prove that $(m,n)(n,k)(k,n)[m,n,k]^2=[m,n][n,k][k,n](m,n,k)^2$

Question

Here (a,b) represents GCD of a and b

[a,b] represents LCM of a and b

I deduced the relation $(m,n,k)^2[m,n,k]=mnk$

But I am not able to proceed further. Any suggestions??

Answer 1

Hint

WLOG the highest exponent of prime $p$ that divides $m,n,k$ be $M,N,K$

WLOG $M\ge N\ge K\ge0$

Now find each LCM and GCD separately

e.g. $[m,n,k]=M$ and $(m,n,k)=K$

This holds true for any prime that divides $mnk$