Remainder

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, 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.

Formally it is also true that a remainder is 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 and the error expression ("the rest") is referred to as the remainder term.

If a and d are integers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < |d|. The number q is called the quotient, while r is called the remainder.

See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder.

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).

At times it is convenient to carry out the division so that a is as close as possible to an integral multiple of d, that is, we can write

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,

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