Let $d=(m,n)$ and $m=da,n=db$. We have \begin{align*} & (m,n)+[m,n]=m+n\\ \iff& d+dab=da+db\\ \iff& 1+ab=a+b\\ \iff& (a-1)(b-1)=0\\ \iff& a=1 \text{ or } b=1\\ \iff& m\mid n \text{ or } n \mid m. \end{align*}