I haven't the faintest idea how to do this. The homework question reads:

Find all positive integer n such that $\displaystyle \varphi(n) = 6$. Briefly explain why.

I know that:

- if $\displaystyle n$ is prime, then $\displaystyle \varphi(n)=n-1$.
- if $\displaystyle n=pq$, where $\displaystyle p$ and $\displaystyle q$ are distinct primes, then $\displaystyle \varphi(n)=(p-1)(q-1)$.
- if $\displaystyle n=p^e$ then $\displaystyle \varphi(n)=p^e-p^{e-1}$.
- if $\displaystyle n=p_1^{e_1}, p_2^{e_2},...p_r^{e_r}$, where $\displaystyle p_i$ are distinct primes, then $\displaystyle \varphi(n)=N(1-1/p_1)(1-1/p_2)...(1-1/p_r)$ (which is really just a generalization of step 3).

I can't find any other info about $\displaystyle phi(n)$ in my notes or in the chapters we've done this semester.

I only come up with the answer $\displaystyle n=7$. But the question wants all values of $\displaystyle n$ which result in $\displaystyle \varphi(n) = 6$.

How do I do this please?

Thanks