Odd Fermat primes proof in an $n$-gon

Question

The regular $n$-gon is constructible by ruler and compass precisely when the odd prime factors of $n$ are distinct Fermat primes. Prove that in this case, $φ(n)$ is a power of $2$. Where $φ(n)$ is Euler's totient function

I think this means $n$ will have distinct factors of the form $2^{2^i}+1$ but can't see where to go from here.

Answer 1

HINT $\phi$ is multaplicative (i.e. $\phi(ab) = \phi(a)\phi(b)$ when $a$ and $b$ are relatively prime) and for a prime $p,$ $\phi(p^k) = p^{k-1}(p-1).$