Definition of notation $\mathbb Z_n$

Question

What does the notation $\mathbb Z_n$ mean, where $n$ is also an integer. I have only seen $n$ being a positive integer up to now. some examples are $\mathbb Z_2$ or $\mathbb Z_3$

This is the context: How to prove $x^{2}+x=1$ has a solution in $\mathbb{Z}_{p}$ if and only if $p=5$ or $p\equiv \pm1\bmod 5$

Answer 1

$\mathbb{Z}_n$ is another (shorter) name for $\mathbb{Z}/n\mathbb{Z}$, the ring of residue classes modulo $n$. A residue class modulo $n$ is the set of all integers which give the same rest when divided by $n$. There are exactly $n$ residue classes, corresponding to the $n$ reminders on division by $n$, $0$ to $n-1$.

The key point is that the reminder of $i+j$ on division by $n$ is the sum of the individual reminders, and the analogue is true for the product. So no matter which member of two given residue classes you choose, the sum will always be in the same residue class, as will be the product.

It only makes sense if $n$ is an integer $\ge 2$. If $n$ happens to be prime, it is a field, that's why those are most interesting. Strictly speaking you could also write $\mathbb{Z}_1$, but that would only contain one element.

Answer 2

$p\equiv\pm1$ (mod 5) means (roughly speaking) that the remainder when dividing $p$ by 5 is 1 or -1. So $p$ could be 4,6,9,11,14, etc. $p$ would still be a positive integer though.