Two integers are called congruent modulo n if their difference is a multiple of n, n being an integer.
We can also say that two integers are congruent modulo n if they have the same remainder from their Euclidean division by 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: \(\equiv\).