What does $c \equiv a \mod N$ mean, and what are equivalence classes mod $N$?

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Accepted Answer

$c \equiv a \mod N$ means that when you divide $c$ by $N$, the remainder is $a$. The mod $N$ operation partitions all integers into $N$ equivalence classes.

* The integers split into 5 residue classes mod 5 — every integer lands in exactly one, giving the finite set $\mathbb{Z}_5 = \{0,1,2,3,4\}$. *
Definition: $c \equiv a \mod N$ iff $N$ divides $(c - a)$, i.e., $c = a + k \cdot N$ for some integer $k$.
Properties:

The result is always in $\{0, 1, 2, ..., N-1\}$
Equivalently: $c \mod N = (a + k \cdot N) \mod N$ for any integer $k$

Equivalence classes mod 5:

$[0] = \{..., -10, -5, 0, 5, 10, 15, ...\}$
$[1] = \{..., -9, -4, 1, 6, 11, 16, ...\}$
$[2] = \{..., -8, -3, 2, 7, 12, 17, ...\}$

The set of equivalence classes mod $N$ is written as $\mathbb{Z}_N = \{0, 1, 2, ..., N-1\}$.
Handling negative numbers: Add multiples of $N$ until positive, then reduce.
Example: $-55 \mod 9 = (-55 + 7 	imes 9) \mod 9 = 8 \mod 9 = 8$
Go deeper:

Modular arithmetic (Wikipedia) — congruence, residue classes and ℤ_N in one place.