Show that $\overline{x}\in\Bbb Z/(n\Bbb Z)$ is invertible iff $\gcd(x,n)=1$

Question

Be $n$ greater than 2 integer and is $\bar x\in\mathbb{Z}_n-\{\bar0,\bar1,\bar2,\ldots,\overline{n-1}\},$ $\;0\leq x<n$. Show that $\exists\; \bar y\in\mathbb{Z}_n$, such that $\bar x\cdot \bar y=\bar y\cdot \bar x=\bar1\Longleftrightarrow gcd\{x,n\}=1$ (i.e., the elements $\bar x$, $0\leq x<n$, invertible in $\mathbb{Z}_n$ são aqueles tais que $gcd\{x,n\}=1$)

Answer 1

By the Bézout's identity:

$$\gcd(x,n)=1\iff \exists y,t\in\Bbb Z,\; yx+tn=1\iff\exists \overline y\in\Bbb Z_n,\; \overline y\overline x=\overline1$$