Prove that a finite cyclic group of order $n>2$ has an even number of distinct generators. What can you deduce about $\phi(n)$ when $n>2$?

Question

I have not yet learned any formulas to compute $\phi(n)$ (Euler phi function), nor am I familiar with its properties. As such, I currently do not have the tools to prove directly that $\phi(n)$ is even for all $n>2$. Is there another way I can prove this? I've been given the hint to make use of the property that the generators occur in pairs, but I'm not sure how to proceed.

I'll admit that I'm a bit lost in this problem. Thanks in advance.

Answer 1

Hint: You may as well assume the group is $\mathbb{Z}_n$. What do you know know about which integers are generators of $\mathbb{Z}_n$? Combine that with your result from the first part about generators coming in pairs.