Show that $U(n)$ is the set of units of the Ring $Z_n$

Question

So in my book the definition of $U(n)$ is the set of all numbers relatively prime to $n$. Later, we find out that this is actually also the set of units of $Z_n$. How would one prove that? What we need to show is That $a \in Z_n$ has a multiplicative inverse $\iff$ $a$ is relatively prime to $n$. I am at loss as to how to do this, any hints?

Answer 1

If $a$ is not relatively prime to $n$, then let $d$ be a common divisor of $a$ and $n$, with $d>1$. Then, for every integer $b$, $d$ divides both $ab$ and $n$ and therefore you can't have $ab\equiv1\pmod n$.

On the other hand, if $\gcd(a,n)=1$ then there an $\alpha\beta\in\mathbb Z$ such that $\alpha a+\beta n=1$ and therefore $\alpha a\equiv 1\pmod n$.