This article is about division of integers. For polynomials, see Euclidean division of polynomials. For other domains, see Euclidean domain.
17 is divided into 3 groups of 5, with 2 as leftover. Here, the dividend is 17, the divisor is 3, the quotient is 5, and the remainder is 2 (which is strictly smaller than the divisor 3), or more symbolically, 17 = (3 × 5) + 2.
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,[1] and modular arithmetic, for which only remainders are considered.[2] The operation consisting of computing only the remainder is called the modulo operation,[3] and is used often in both mathematics and computer science.
The pie has 9 slices, so each of the 4 people receives 2 slices and 1 is left over.
^"Division and Euclidean algorithms". www-groups.mcs.st-andrews.ac.uk. Archived from the original on 2021-05-06. Retrieved 2019-11-15.
^"What is modular arithmetic?". Khan Academy. Retrieved 2019-11-15.
^"Fun With Modular Arithmetic – BetterExplained". betterexplained.com. Retrieved 2019-11-15.
and 26 Related for: Euclidean division information
In arithmetic, Euclideandivision – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way...
two numbers Euclidean domain, a ring in which Euclideandivision may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and...
which allows a suitable generalization of the Euclideandivision of integers. This generalized Euclidean algorithm can be put to many of the same uses...
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...
a modulo n (often abbreviated as a mod n) is the remainder of the Euclideandivision of a by n, where a is the dividend and n is the divisor. For example...
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest...
division (Blomqvist's method). Polynomial long division is an algorithm that implements the Euclideandivision of polynomials, which starting from two polynomials...
Bézout's theorem (named after Étienne Bézout) is an application of Euclideandivision of polynomials. It states that, for every number r , {\displaystyle...
polynomials all the properties that may be deduced from the Euclidean algorithm and Euclideandivision. Moreover, the polynomial GCD has specific properties...
is the Euclidean algorithm, a variant in which the difference of the two numbers a and b is replaced by the remainder of the Euclideandivision (also called...
the result of Euclideandivision. Some are applied by hand, while others are employed by digital circuit designs and software. Division algorithms fall...
synthetic division is a method for manually performing Euclideandivision of polynomials, with less writing and fewer calculations than long division. It is...
and rem uses the sign of the dividend. Euclideandivision of polynomials is very similar to Euclideandivision of integers and leads to polynomial remainders...
many properties with integers: they form a Euclidean domain, and have thus a Euclideandivision and a Euclidean algorithm; this implies unique factorization...
remainders of the Euclideandivision of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product...
there is a notion of Euclideandivision of polynomials, generalizing the Euclideandivision of integers. This notion of the division a(x)/b(x) results in...
the Euclideandivision by P of the product in GF(p)[X]. The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm;...
part of a division (in the case of Euclideandivision) or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the...
way of Dichotomy or Bipartition being the most natural and easie kind of Division, that Number is capable of this down to an Unite". In 1716, King Charles...
uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons, and subtraction....
and get a natural number as result, the procedure of division with remainder or Euclideandivision is available as a substitute: for any two natural numbers...
remainder of the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclideandivisions. The above says...
Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests...
for very large integers n, one can use the quotient of Euclideandivision for both of the division operations. This has the advantage of only using integers...
algorithms Trachtenberg developed are ones for general multiplication, division and addition. Also, the Trachtenberg system includes some specialised methods...
of Euclideandivision, the remainder would be included as well. Using short division, arbitrarily large dividends can be handled. Short division does...
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