Prove that for any integers $m$ and $n$ we have to: $mn=(m,\,n)[m,\,n]$.

Question

Prove that for any integers $m$ and $n$ we have to: $mn=(m,\,n)[m,\,n]$ ($(m,\,n)$ being the gcd & $[m,\,n]$ the lcm). How can i prove it?

Answer 1

Write prime factorizations $m=\prod_ip_i^{a_i},\,n=\prod_ip_i^{b_i}$ over all primes that divide at least one of $m,\,n$. The problem reduces to $a_i+b_i=\min\{a_i,\,b_i\}+\max\{a_i,\,b_i\}$.

Answer 2

Hint:

Let $d=\gcd(m,n)$ so that $m=m'd$, $\:n=n'd\:$ and $\:m',n'$ are coprime. Then $\operatorname{lcm}(m,n)=m'n=mn'$, so that $$mn=m'd\,n'd=\dots$$