Two numbers \( a, b \in \mathbb{Z} \) are said to be congruents module p if

$$a-b = np, n \in \mathbb{Z} . $$
In oher words, a and b are congruents if a minus b is a multiple of m.

When a is congruent with b module p, we denote it as

$$ a \equiv b (p) . $$
Also, sometimes it is write as
$$ a \equiv_{p} b. $$

Examples

• We want to take p = 9, a = 3, b = 8 then $$a – b = 3 – 8 = -5 $$ so, 3 is not congruent with 8 module 9.
•  \(13 \equiv 4 (9)\) because \(13-4 = 9 \).

Congruences and equivalence classes
Congruences gives an equivalence relationship, so, original set is splitted in equivalence classes. We define it as

$$ \mathbb{Z}_{p} = \frac{\mathbb{Z}}{\equiv_{p}} . $$

For example, the set \(\mathbb{Z} \) with p=3, the number 19 belongs to \( \{ \bar{1}\} \) equivalence class of \( \mathbb{Z}_{3}\) because
$$ 19 – 1 = 18 = 6 . 3. $$

For example when p = 9, \( \mathbb{Z}_{9} = \{ \bar{0},\bar{1},\bar{2},\bar{3},\bar{4},\bar{5},\bar{6},\bar{7},\bar{8} \} \), the equivalence class \({\bar{2}}\) is formed for those all numbers such us when we divide if by 9 gives as rest 2 i.e the set of numbers \(\{ 2, 11, 20, …\} \).

In general \( \mathbb{Z}_{p} = \{ \bar{0},\bar{1},\bar{2}, …\bar{p-1} \} \).

Congruences and big structures

Congruences and group Theory

What makes interesting congruences is the set \(\mathbb{Z}_{p}\) dotted of sum ‘+’ operation has a group structure, it is denoted by

$$ (\mathbb{Z}_{p}, +) $$

Example the \( (\mathbb{Z}_{3}, +) \) group

Congruencias y teoría de Anillos
También \(\mathbb{Z}_{p}\) dotado de la suma ‘+’ y la multiplicación ‘.’ tiene estructura de anillo, y lo denotamos por
$$ (\mathbb{Z}_{p}, +, .). $$

El caso en que p es un número primo tiene especial importante, porque todo elemento no nulo de \(\mathbb{Z}_{p}\) tiene elemento inverso para el producto ‘.’. y así el anillo $$ (\mathbb{Z}_{p}, +, .). $$ tiene en realidad estructura de cuerpo.