divisies

Divisies is a term used in mathematics to refer to two closely related ideas: the operation of division and the property of divisibility. Division is the process of splitting a quantity into equal parts. For integers a and nonzero b, the division of a by b yields a quotient q and, if a is not exactly divisible, a remainder r, such that a = b q + r with 0 ≤ r < |b|. If r = 0, then b is a divisor of a and a is divisible by b.

Divisibility is a relation between integers. We say that b divides a if there exists an integer q with a = b q. In this sense, every integer has a set of divisors, and the corresponding set of multiples. The greatest common divisor of two integers is the largest integer that divides both, and is found efficiently by the Euclidean algorithm. Modular arithmetic studies remainders and equivalence classes modulo a number, a central tool in number theory and cryptography.

In algebra, division also applies to polynomials. Polynomial division produces a quotient and a remainder with degree of the remainder smaller than the divisor, analogous to integer division. The concepts extend to rings and fields, and to more general objects in abstract algebra.

Important caveats include that division by zero is undefined, and that division is not always exact in integers unless the remainder is zero. Divisies thus covers both the procedure of dividing quantities and the property of exact divisibility that underpins many results in mathematics.