Construction of regular $e$-gon with ruler and compass.

Question

Let $n,m\in \mathbb{Z}_{>0}$ and $e=lcm(m,n)$. We assume that regular $n-$gon and regular $m-$gon are constructive with ruler and compass. Show that regular $e-$gon are constructive with ruler and compass.
We khow that regular $n$-gon is constructive with ruler and compass iff $\phi(n)=2^{k}$, $k\in \mathbb{N}$.So we have that $\phi(n)=2^{a},\ \phi(m)=2^{b},\ a,b\in \mathbb{N}$. How can we proof that $\phi(e)$ is power of $2$?

Answer 1

HINT: Express $\varphi(n)$ and $\varphi(m)$ in terms of the prime factorizations of $n$ and $m$ and see what this tells you about the primes involved.