In modular arithmetic, numbers “wrap around” when they reach a fixed quantity. This is also called the modulus—thus the name modular arithmetic—with the standard way of writing the form as “mod 12” or “mod 2.”

In this case, if the two numbers b (also called the base) and c (also called the remainder) are subtracted (b - c), and their difference is a number integrally divisible by m, or (b - c)/m, then b and c are said to be congruent modulo m. Mathematically, “b is congruent to c (modulo m)” is written as follows, with the symbol for congruence (≡):

b ≡ c (mod m)

But if b - c is not integrally divisible by m, then it is said, “b is not congruent to c (modulo m),” or

b ≢ c (mod m)

More formally, modular arithmetic includes any “non-trivial homomorphic image of the ring of integers.” We can interpret this interesting definition using a clock. The modulus would be the number 12 on the clock (arithmetic modulo 12), with an associated ring labeled C₁₂ and the allowable numbers being 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. Another example is arithmetic modulo 2, with an associated ring of C₂, or allowable numbers of 1 and 2.