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Concept
Euclidean division
Summary
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder exist and are unique, under some conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division.
Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the modulo operation, and is used often in both mathematics and computer science.
Euclidean division is based on the following result, which is sometimes called Euclid's division lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r
and
0 ≤ r < b,
where b denotes the absolute value of b.
In the above theorem, each of the four integers has a name of its own: a is called the , b is called the , q is called the and r is called the .
The computation of the quotient and the remainder from the dividend and the divisor is called , or in case of ambiguity, . The theorem is frequently referred to as the (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing q and r (see the section Proof for more).
Division is not defined in the case where b = 0; see division by zero.
For the remainder and the modulo operation, there are conventions other than 0 ≤ r < b, see .
Official source
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