# Euclidean division

Division theoremdividedivisible by two without remainderdivisiondivision algorithmdivision algorithmsdivision with remainderEuclidean division theoremGeneralized division algorithmsinteger division
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.wikipedia
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### Division (mathematics)

divisiondividingdivided
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.
The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated.

### Division algorithm

Newton–Raphson divisionSRT divisiondivision by a constant
The methods of computation are called integer division algorithms, the best known of which being long division. Although "Euclidean division" is named after Euclid, it seems that he did not know the existence and uniqueness theorem, and that the only computation method that he knew was the division by repeated subtraction.
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division.

### Modulo operation

modulomodmodulus
The operation consisting of computing only the remainder is called the modulo operation, and is used often in both mathematics and computer science.
) is the remainder of the Euclidean division of

### Modular arithmetic

modulomodcongruent
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.
explicitly showing its relationship with Euclidean division.

### Remainder

in remainderlaver
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.
See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder.

### Greatest common divisor

gcdcommon factorgreatest common factor
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.
When Lehmer's algorithm encounters a too large quotient, it must fall back to one iteration of Euclidean algorithm, with a Euclidean division of large numbers.

### Euclidean algorithm

Euclid's algorithmEuclideanEuclid
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 theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique.

### Integer

integersintegralZ
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.
It is called Euclidean division and possesses the following important property: that is, given two integers

### Arithmetic

arithmetic operationsarithmeticsarithmetic operation
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.
Within natural numbers there is also a different, but related notion, the Euclidean division, giving two results of "dividing" a natural N (numerator) by a natural D (denominator), first, a natural Q (quotient) and second, a natural R (remainder), such that

### Euclidean domain

EuclideanEuclidean functionDomain
Euclidean domains (also known as Euclidean rings) are defined as integral domains which support the following generalization of Euclidean division:
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers.

### Modular multiplicative inverse

multiplicative inversemodular inverseinverse mod
Given integers a, m and R, with m >0 and let R^{-1} be the modular multiplicative inverse of R (i.e., 0
The division algorithm shows that the set of integers,

### Polynomial

polynomial functionpolynomialsmultivariate polynomial
It occurs only in exceptional cases, typically for univariate polynomials, and for integers, if the further condition
The Euclidean division of polynomials that generalizes the Euclidean division of the integers.

### Gaussian integer

Gaussian integersGaussian primeGaussian primes
Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers.
Gaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials.

### Montgomery modular multiplication

Montgomery reductionMontgomery multiplicationMontgomery multiplier
-residue defined in Montgomery reduction.
can be expressed by applying the Euclidean division theorem:

### Polynomial greatest common divisor

greatest common divisorEuclidean divisionEuclidean division of polynomials
See Polynomial long division, Polynomial greatest common divisor#Euclidean division and Polynomial greatest common divisor#Pseudo-remainder sequences.
The similarity between the integer GCD and the polynomial GCD allows us to extend to univariate polynomials all the properties that may be deduced from Euclid's algorithm and Euclidean division.

### Natural number

natural numberspositive integerpositive integers
Other proofs use the well-ordering principle (i.e., the assertion that every non-empty set of non-negative integers has a smallest element) to make the reasoning simpler, but have the disadvantage of not providing directly an algorithm for solving the division (see for more).
. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

### Quotient

In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.

### Long division

(long)Division tableauSchoolbook long division
The methods of computation are called integer division algorithms, the best known of which being long division.

### Uniqueness quantification

uniqueuniquenessone and only one
, there exist unique integers

### Absolute value

modulusabsolutemagnitude
denotes the absolute value of

### Division by zero

divide by zerodividing by zerodivided by zero
; see division by zero.

### Euclid

Euclid of AlexandriaEuklidGreek Mathematician
Although "Euclidean division" is named after Euclid, it seems that he did not know the existence and uniqueness theorem, and that the only computation method that he knew was the division by repeated subtraction.

### Hindu–Arabic numeral system

Hindu-Arabic numeral systemHindu-Arabic numeralsHindu numerals
Before the discovery of Hindu–Arabic numeral system, which was introduced in Europe during the 13th century by Fibonacci, division was extremely difficult, and only the best mathematicians were able to do it.

### Fibonacci

Leonardo FibonacciLeonardo of PisaLeonardo Pisano
Before the discovery of Hindu–Arabic numeral system, which was introduced in Europe during the 13th century by Fibonacci, division was extremely difficult, and only the best mathematicians were able to do it.

### Divisor

divisibilitydividesdivisible
As an alternate example, if 9 slices were divided among 3 people instead of 4, then each would receive 3 and no slice would be left over, which means that the remainder would be zero, leading to the conclusion that 3 evenly divides 9, or that 3 divides 9.