Let be $W_n$ and $W_n$ two complex numbers with respectively $n$-th and $m$-th roots, is clear that $W_nW_m$ is a $k$-th root for the unity. What is the less integer value for $k$? Why?
My candidate is $k=mcm(m,n)$, let be $A=${$r\in Z+$: $(W_nW_m)^{r}=1$} A is not empty cause $nm\in A$ and $A\subset N$ then there exist a $d:=min(A)$ Is k=d?