Notation

Definition 1.1 (Congruence)   Let $a,b\in\mathbb{Z}$ and $n\in\mathbb{N}$. Then

$$a\equiv b\pmod{n}$$

if $n\mid a-b$.

That is, there is $c\in\mathbb{Z}$ such that

$$nc = a-b.$$

One way I think about it: $a$ is congruent to $b$ modulo $n$, if we can get from $b$ to $a$ by adding multiples of $n$.

Congruence modulo $n$ is an equivalence relation. Let

$$\mathbb{Z}/n\mathbb{Z}= \{$$ the set of equivalence classes $$\}$$

The set $\mathbb{Z}/n\mathbb{Z}$ is a ring, the ``ring of integers modulo $n$''. It is the quotient of the ring $\mathbb{Z}$ by the ideal generated by $n$.

Example 1.2  

$$\mathbb{Z}/3\mathbb{Z}= \{ \{\ldots, -3, 0, 3, \ldots\}, \{\ldots, -2, 1, 4, \ldots\}, \{\ldots, -1, 2, 5, \ldots\}\} = \{[0], [1], [2]\}$$

where we let $[a]$ denote the equivalence class of $a$.