I am a high school student self-studying Number Theory and came across this question in the book Challenge and Thrill of Pre-College Mathematics (For reference, $(m,n)$ means $\gcd(m,n)$ and $[m,n]$ means $\text{LCM}(m,n)$):
If $m$, $n$, and $k$ are any three positive integers, prove that
$$(m,n)(m,k)(n,k)[m,n,k]^2=[m,n][m,k][n,k](m,n,k)^2$$
I was able to derive this identity by trial and error and then prove it mathematically:
$$\frac{(m,n)(m,k)(n,k)[m,n,k]}{(m,n,k)}=mnk$$
And I suspect that this may be true as well:
$$\frac{[m,n][m,k][n,k](m,n,k)}{[m,n,k]}=mnk$$
Which would prove the proposition. However, I am unable to prove this statement and am unsure of its truth.
Please offer a hint to how I would go about proving the second part, or in case if my assumption is incorrect than the correct method of proof.
We can use that $[m,n]\cdot(m,n)=m\cdot n\Rightarrow [m,n]=\frac{mn}{(m,n)}$.
So
$$\frac{[m,n][m,k][n,k](m,n,k)}{[m,n,k]}=\frac{mn\cdot mk\cdot nk \cdot(m,n,k)}{(m,n)(m,k)(n,k)[m,n,k]} = m^2n^2k^2 \cdot A $$
Where $A$ is the inverse of the term you calculated to be $mnk$.