Hey guys, I need some help here.

a)Let $\displaystyle n$ and $\displaystyle a$ be positive integers and $\displaystyle p$ be a prime. Show $\displaystyle (p-1) | \varphi(n)$ and $\displaystyle p^{a-1} | \varphi(n)$ given $\displaystyle p^a | n$.

b) Using part a, find all positive integers n for $\displaystyle \varphi(n) =12$.

a)

Since $\displaystyle p^a | n$, this implies $\displaystyle \varphi(p^a) | \varphi(n)$

And $\displaystyle \varphi(p^a)=p^a-p^{a-1}=(p-1)p^{a-1}$

Thus $\displaystyle (p-1) | \varphi(n)$ and $\displaystyle p^{a-1} | \varphi(n)$.

Is this good enough?

b) I am stuck on this one.

Thanks for your help.