TITLE: Find $n\in \mathbb N$ so that $(\mathbb Z_n, +, \cdot)$ has exactly 4 invertible elements and 5 zero-divisors
QUESTION [5 upvotes]: Find $n\in \mathbb N$ so that $(\mathbb Z_n, +, \cdot)$ has exactly 4 invertible elements and 5 zero-divisors. 
As I couldn't find any theorem that would lead me toward a solution, so  I have been trying guessing and checking with no results so far, so I am asking for a push in the right direction.

REPLY [5 votes]: HINT: $x$ is a unit in $\mathbb{Z}_n$ if and only if gcd$(n,x) = 1$
EDIT: furthermore: Let $R$ be a finite ring with unity, and $x \in R$: $x$ is a unit in $R$ if and only if $x$ is no zero divisor and $x \neq 0$