Two integers are called

**congruent modulo**if their difference is a multiple of*n**n, n*being an integer.We can also say that two integers are

**congruent modulo**if they have the same remainder from their Euclidean division by*n**n*.In modular arithmetic modulo *n*, the results of operations are expressed modulo *n*.

### Examples

The numbers 15 and 3 are congruent modulo 12. In fact, their difference, 12, is a multiple of 12.

Also, the remainder of the division of 15 and 3 by 12 is 3 in both cases.

15 ÷ 12 = 1 × 12 + 3

3 ÷ 12 = 0 × 12 + 3

\(\frac{15}{3}\) and \(\frac{3}{12}\)

15 and 3 have the same remainder.

In arithmetic modulo 5, we can write: 3 + 4 = 2, because 3 + 4 = 7 and 7 is congruent to 2 modulo 5.

We can also write: 3 + 4 \(\equiv\) 2 modulo 5.

### Notation

The relationship of congruence modulo n is noted with the symbol: \(\equiv\).