remainder

A remainder is the amount left over after dividing one number by another. For integers a and b with b ≠ 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < |b|. Here q is the quotient and r is the remainder. This form is known as Euclidean division.

Remainders are used to measure divisibility and to perform modular arithmetic. In modular arithmetic, r is the representative of the residue of a modulo b, written a ≡ r (mod b). The remainder determines cyclic patterns, congruences, and computations that depend only on the division remainder rather than the full value of a.

In the division of polynomials, dividing f(x) by a nonzero polynomial d(x) yields f(x) = q(x)d(x) + r(x), where r(x) has degree less than that of d(x). The remainder r(x) is unique and plays a role analogous to the integer remainder in arithmetic.

In computing and programming, the remainder is obtained by the modulo operation, often denoted a mod b. Different programming languages may adopt different conventions for the sign of the remainder when a is negative, but the underlying concept remains the amount left over after division.

Example: 17 divided by 5 gives a quotient of 3 and a remainder of 2, since 17 = 5×3 + 2. For negative dividends, conventions vary; under the common nonnegative remainder convention, -17 = 5×(-4) + 3, so the remainder is 3.

A related concept is the remainder theorem in algebra: the remainder of f(x) divided by x − c is f(c).