remainders

A remainder is the nonnegative value left over after performing division of one integer by another. If integers a and b satisfy 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. For example, dividing 23 by 7 gives q = 3 and r = 2, since 23 = 7×3 + 2.

The remainder is defined by the division algorithm and is unique for a given pair (a, b). For a positive divisor b > 0, the remainder always lies between 0 and b−1. If the dividend a is negative, the quotient may be negative, but the remainder remains nonnegative. When the divisor is negative, the usual convention is to bound the remainder by the absolute value |b|, so 0 ≤ r < |b| still holds.

Remainders have important applications in modular arithmetic. The remainder r obtained from dividing a by b is congruent to a modulo b, written a ≡ r (mod b). This concept underpins computations of residues, cyclic structures, and algorithms for greatest common divisors, such as the Euclidean algorithm, which repeatedly uses remainders to reduce numbers.

In algebra, remainders also appear in polynomial division: when a polynomial f(x) is divided by x − c, the remainder is f(c) (the Remainder Theorem). In numeral systems, remainders play a role in digit extraction and base representations, where successive divisions by the base reveal the digits of a number.