When is $U(n)$ a cyclic group?

Question

We have been taught $U(n)=\{k: 1\le k \le n-1, (k,n)=1\}$.

I am stuck in proving the idea that every element has an inverse in the group.Some help is appreciated since I am not able to start the problem.

I also wanted to know when is this group $U(n)$ a cyclic group?Is there a theorem for it?

Answer 1

Hint:-What does the definition of $\gcd$ tell you?

If $\gcd(a,n)=1$ , then there exists integers $u$ and $v$ such that $au+nv=1$. What does that tell you if you reduce the above equation modulo $n$ ? . Do you find your required inverse?.