TITLE: Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
QUESTION [8 upvotes]: I know that there is a similar post, but I 'm trying a different proof.
I will write $P$  for the set of all positive prime numbers.
Question: If $\phi$ is Euler's Phi Function, we want to find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$.
Answer: Let $n=p_1^{n_1}\dotsb p_k^{n_k}\in \mathbb{Z}^+$ be the factorisation of $n$ in to primes. Then
$$\phi(n)=p_1^{n_1-1} \dotsb p_k^{n_k-1}(p_1-1) \dotsb (p_k-1)=4.$$
So, for any $i \in \{1,2,\cdots,k\}$ we have $p_i-1|4$. Hence,
$$p_i-1\in\{1,2,4\} \iff p_i\in \{2,3,5\} \subset P.$$
Now, we can see the primes that $n$ contains: $n=2^{n_1}3^{n_2}5^{n_3}$, where  $n_1,n_2,n_3 \in \mathbb{Z}^+$. So,
\begin{align*}
\phi(2^{n_1}3^{n_2}5^{n_3})=4 \iff \phi(2^{n_1})\phi(3^{n_2})\phi(5^{n_3})=4  \tag{*}
\end{align*}
The possible cases for $n_i$ are:

$n_1=1,2,3\implies \phi(2)=1,\phi(2^2)=2, \phi(2^3)=4$ respectively
$n_2=1 \implies \phi(3)=2$
$n_3=1 \implies \phi(5)=4$

All the posible combinations for the relation (*) are $\phi(5)$, $\phi(5)\phi(2)$, $\phi(3)\phi(2^2)$, $\phi(2^3)$. So, $$n \in \{5,10,12,8\}.$$
Is this completely right?
Thank you.

REPLY [3 votes]: This seems to be completely correct to me.