Let $p$ be a prime and $\mathbb{Z}_p=\{0,1,2,\dots,p-1\}$.
"Any $x\in \mathbb{Z}_p$ except $0$, has a modulo multiplicative inverse."
My question is:
In definition, there is a $y$ that $xy ≡ 1 \pmod{p}$. $y$ must be an integer in $\mathbb{Z}_p$, or $y$ can be any integers?
If $x$ is non divisible by $p$, we can find infinite $y\in\mathbb{Z}$ such that $xy\equiv 1 \pmod{p}$. They are all equivalent modulo $p$, that is if $xy_1\equiv 1 \pmod{p}$ and $xy_2\equiv 1 \pmod{p}$, then $y_1\equiv y_2 \pmod{p}$. Exactly one of them, say $y_0$, is in the set $\{1,2,\dots,p-1\}$ and it can be considered as the representative of this equivalent class: any $y\in\mathbb{Z}$ such that $xy\equiv 1 \pmod{p}$ can be written as $y=y_0+kp$ for some $k\in\mathbb{Z}$.