# Generators of a cyclic group; how do you calculate them

The question reads: Find the number of generators of a cyclic group having the given order.

A) 8

B) 60

This is a practice question for the quiz. The answers are 4 and 16, but I'm confused how it's calculated. Okay, so the cyclic group has a cardinality of 8. Do I need to figure out how many generators produce Z8? Z8 isn't mentioned, but I don't know what else to do.

• Just a note. $\mathbb{Z}8$ is the only cyclic group of order 8. – baru Oct 21 '16 at 2:33

Recall Euler's $\phi$ function: $\phi (n)=1$ if $n=1$ and $\phi (n)$ is the number of integers $1\leq k \leq n-1$ such that $(n,k)=1$ for $n>1$. Prove that if $G$ is a cyclic group with order $n$, then the number of generators of $G$ is $\phi (n)$.