Let $z \in \mathbb{Z}$ a number with characteristic
$$ \exists x,y \in \mathbb{Z} \: 1 = xz + yn $$
Show that $[z]_n$ in $(\mathbb{Z}_n, *)$ is invertibel. Figure out invertibel element also.
$* = $ Multiplication operation.
My proof:
Using $\exists x,y \in \mathbb{Z} \: 1 = xz + yn$, using Euclidean algorithm it is true that $ggT(z,n) = 1$.
$\Rightarrow 1 \equiv xz \mod n$
So is $x \mod n$ inverse to z.
Does this proof suffice?