Modulo

Modulo refers to the operation of finding the remainder after division of one number by another. In the standard mathematical convention, for integers a and n with n > 0, a mod n is the unique remainder r such that a = qn + r for some integer q and 0 ≤ r < n. The value of a mod n is called the residue of a modulo n.

In algebra, modulo n is closely linked to the concept of congruence. Two integers a and b are congruent modulo n, written a ≡ b (mod n), if n divides their difference a − b. The residue class of a modulo n consists of all integers that are congruent to a modulo n, and a mod n is a canonical representative of that class.

Modulo arithmetic is closed under addition, subtraction, and multiplication. That is, (a + b) mod n = ((a mod n) + (b mod n)) mod n, and similarly for subtraction and multiplication. Division is not always defined in modular arithmetic; inverses exist only for elements coprime to n, leading to the concept of modular inverses and the multiplicative group of units modulo n. The set of integers modulo n forms a finite ring, and when n is prime, it forms a field with respect to addition and multiplication.

Common uses include solving problems in number theory, cryptography (such as RSA and elliptic-curve systems), calendar calculations, hashing, and algorithms that rely on cyclic properties and modular residue reduction. The terms modulo, modulus, and remainder are related but can be used in slightly different senses: modulo often denotes the equivalence relation, while remainder refers to the numerical residue from division. In programming, the exact behavior of a mod n may vary by language, especially for negative inputs.