Definition:Congruence (Number Theory)

General Definition
Let $$z \in \R$$.

Definition by Equivalence Relation
We define a relation $$\mathcal{R}_z$$ on the set of all $$x, y \in \R$$:
 * $$\mathcal{R}_z = \left\{{\left({x, y}\right) \in \R \times \R: \exists k \in \Z: x = y + k z}\right\}$$

This relation is called congruence modulo $$z$$, and the real number $$z$$ is called the modulus.

If $$\left({x, y}\right) \in \mathcal{R}_z$$, we write:
 * $$x \equiv y \left({\bmod\, z}\right)$$

and say "$$x$$ is congruent to $$y$$ modulo $$z$$."

Similarly, if $$\left({x, y}\right) \notin \mathcal{R}_z$$, we write:
 * $$z \not \equiv y \left({\bmod\, z}\right)$$

and say "$$x$$ is not congruent (or incongruent) to $$y$$ modulo $$z$$."

We have that congruence modulo $z$ is an equivalence relation.

Definition by Modulo Operation
Let $$z \in \R: z \ne 0$$ be defined as the modulo operation.

Then:
 * $$x \equiv y \left({\bmod\, z}\right) \iff x \,\bmod\, z = y \,\bmod\, z$$

Definition by Integral Multiple
Equivalently, $$x$$ is congruent to $$y$$ modulo $$z$$ iff their difference is an integral multiple of $$z$$:
 * $$x \equiv y \left({\bmod\, z}\right) \iff \exists k \in \Z: x - y = k z$$

Definition for Integers
The concept of congruence is usually considered in the integer domain.

Let $$m \in \Z, m > 0$$.

Then we define congruence modulo $$m$$ as the relation $$\mathcal{R}_m$$ on the set of all $$a, b \in \Z$$:
 * $$\mathcal{R}_m = \left\{{\left({a, b}\right) \in \Z \times \Z: \exists k \in \Z: a = b + km}\right\}$$

The other definitions also apply under the same restriction.

Thus we see that $$a$$ is congruent to $$b$$ modulo $$m$$ if their difference is a multiple of $$m$$:


 * $$a \equiv b \left({\bmod\, m}\right) \iff m \backslash \left({a - b}\right)$$

This gives us an alternative method of defining congruence modulo an integer.

Equivalence of Definitions
The definitions as given here are equivalent.

Residue
A residue of $$a$$ modulo $$z$$ is another word meaning remainder, and means any number congruent to $$a$$ modulo $$z$$.