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Concept
Remainder
Summary
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation is the operation that produces such a remainder when given a dividend and divisor.
Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the difference. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion, where the error expression ("the rest") is referred to as the remainder term.
Given an integer a and a non-zero integer d, it can be shown that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < . The number q is called the quotient, while r is called the remainder.
(For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see division algorithm.)
The remainder, as defined above, is called the least positive remainder or simply the remainder. The integer a is either a multiple of d, or lies in the interval between consecutive multiples of d, namely, q⋅d and (q + 1)d (for positive q).
In some occasions, it is convenient to carry out the division so that a is as close to an integral multiple of d as possible, that is, we can write
a = k⋅d + s, with |s| ≤ |d/2| for some integer k.
In this case, s is called the least absolute remainder. As with the quotient and remainder, k and s are uniquely determined, except in the case where d = 2n and s = ± n. For this exception, we have:
a = k⋅d + n = (k + 1)d − n.
A unique remainder can be obtained in this case by some convention—such as always taking the positive value of s.
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