The concept of numerical congruence was formulated by Gauss. If two numbers have the same remainder when divided by a given number m (called the modulus), then they are said to be congruent modulo m. We say "a is congruent to b (modulo m)" or, symbolically, a ≡ b (mod m). Another way of defining the concept is to say that two numbers are congruent modulo m if m divides the difference of the two numbers (we usually assume that the modulus is greater than zero). For example, 18 and 25 both leave a remainder of 4 when divided by 7 and are therefore congruent modulo 7.
Modular arithmetic is occasionally referred to as "clock arithmetic." If it is 11:00 right now, two hours later it will be 1:00, and 12 hours later it will be 11:00 again. Adding and subtracting time on a clock is quite similar to modular arithmetic modulo 12, although in modular arithmetic the numbers would range from 0 to 11, not 1 to 12.
Here is a sample of recreational applications of congruences:
There is also a congruences worksheet available.