Find the number of generators of the cyclic group $\mathbb{Z}_{p^r}$ Let $p$ be a prime number. Find the number of generators of the cyclic group $\mathbb{Z}_{p^r}$, where $r \in \mathbb{Z} \geq 1$
I'm trying to understand the question and am experimenting with $p=5$ and $r=1,2,3$.
When $r=1$ it generates $\mathbb{Z_5}$, where every non-zero element is a generator of the group.
When $r=2$ it generates $\mathbb{Z_{10}}$. All the elements relatively prime to $10$ are $1,3,7,$ and $9$, also $4$ generators.
When $r=3$ it generates $\mathbb{Z_{15}}$. All of the elements relatively prime to $15$ are $1,2,4,7,8,11,13$, and $14$, which are $8$ generators.
So I'm trying to figure out how to find the number of relatively prime elements for the general group $\mathbb{Z}_{p^r}$ 

I personally would use the prime counting function as follow: $ π(p^r)−π(1)$, with $π(x)=π(x)∼ \frac{x}{log(x)}$, so the number of generetor for the cyclic group $\mathbb{Z}_{p^r}$ is given by $z(p^r)=π(p^r)−π(1)$.
