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 $$z \in \R$$ 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 Zero
Note that when $$z = 0$$ the definition is still meaningful:
 * $$x \equiv y \left({\bmod\, 0}\right) \iff x \,\bmod\, 0 = y \,\bmod\, 0 \iff x = y$$

and:
 * $$x \equiv y \left({\bmod\, 0}\right) \iff \exists k \in \Z: x - y = 0 \cdot k = 0 \iff x = y$$

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 $$z$$.

Historical Note
The concept of congruence modulo an integer was first explored by Carl Friedrich Gauss.

Linguistic Note
The word modulo comes from the Latin for with modulus, that is, with measure.