How do congruence classes group integers with the same remainder?

In number theory, congruence classes—also known as residue classes—partition the set of all integers into groups based on their remainders when divided by a fixed modulus $N$.

Two integers $a$ and $b$ belong to the same congruence class modulo $N$ if their difference, $a-b$, is exactly divisible by $N$. This relationship is expressed as $a\equiv b\left(\mathrm{mod}N\right)$. Because any integer in a class can represent that class, the group effectively collects all numbers that leave the same remainder upon division by $N$. For example, with a modulus of 5, the integers $\dots ,-6,-1,4,9,\dots$ all share the same remainder and thus form a single congruence class.12

This system is fundamental to modular arithmetic, allowing mathematicians to perform operations like addition, subtraction, and multiplication on these classes by choosing any representative member from each group to compute the result.1