In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4.[1][2]
In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc.[3][4][5][6] Historically, other names for the same concept have included greatest common measure.[7]
This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see § In commutative rings below).
^Long (1972, p. 33)
^Pettofrezzo & Byrkit (1970, p. 34)
^
Kelley, W. Michael (2004), The Complete Idiot's Guide to Algebra, Penguin, p. 142, ISBN 978-1-59257-161-1.
^Jones, Allyn (1999), Whole Numbers, Decimals, Percentages and Fractions Year 7, Pascal Press, p. 16, ISBN 978-1-86441-378-6.
^Cite error: The named reference Hardy&Wright 1979 20 was invoked but never defined (see the help page).
^Some authors treat greatest common denominator as synonymous with greatest common divisor. This contradicts the common meaning of the words that are used, as denominator refers to fractions, and two fractions do not have any greatest common denominator (if two fractions have the same denominator, one obtains a greater common denominator by multiplying all numerators and denominators by the same integer).
^
Barlow, Peter; Peacock, George; Lardner, Dionysius; Airy, Sir George Biddell; Hamilton, H. P.; Levy, A.; De Morgan, Augustus; Mosley, Henry (1847), Encyclopaedia of Pure Mathematics, R. Griffin and Co., p. 589.
and 23 Related for: Greatest common divisor information
In mathematics, the greatestcommondivisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of...
several ways to compute least common multiples. The least common multiple can be computed from the greatestcommondivisor (gcd) with the formula lcm ...
extension to the Euclidean algorithm, and computes, in addition to the greatestcommondivisor (gcd) of integers a and b, also the coefficients of Bézout's identity...
or Euclid's algorithm, is an efficient method for computing the greatestcommondivisor (GCD) of two integers (numbers), the largest number that divides...
properties such as the existence of a Euclidean algorithm for computing greatestcommondivisors, Bézout's identity, the principal ideal property, Euclid's lemma...
In mathematics, a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,} is an integer m {\displaystyle m} that may...
lowest common multiple of the denominators of a set of fractions Greatestcommondivisor, the largest positive integer that divides each of the integers...
particularly ring theory, maximal commondivisors are an abstraction of the number theory concept of greatestcommondivisor (GCD). This definition is slightly...
expressing this fact in mathematical notation is to indicate that their greatestcommondivisor is one, by the formula gcd(a, b) = 1 or (a, b) = 1. In their 1989...
quotients of a and b (respectively) by the greatestcommondivisor of a and b. Proof: If d is this greatestcommondivisor, Bézout's identity asserts the existence...
if and only if a and b are coprime, that is, if a and b have a greatestcommondivisor of 1. In higher mathematics, "irreducible fraction" may also refer...
binary Euclidean algorithm, is an algorithm that computes the greatestcommondivisor (GCD) of two nonnegative integers. Stein's algorithm uses simpler...
phrase least commondivisor is a confusion of the following two distinct concepts in arithmetic: Least common multiple Greatestcommondivisor This disambiguation...
generally, with coefficients in a unique factorization domain) is the greatestcommondivisor of its coefficients. The primitive part of such a polynomial is...
those of f, and the greatestcommondivisor of two polynomials is independent of the ambient field, so the greatestcommondivisor of f and f′ has coefficients...
CA]. Tomas Garcia, Rogelio (August 2020). "Equalities between greatestcommondivisors involving three coprime pairs" (PDF). Notes on Number Theory and...
concerning integers, such as the Euclidean algorithm for finding the greatestcommondivisor of two integers, and modular arithmetic, for which only remainders...
provide a geometrical method to determine the least common multiple and the greatestcommondivisor of two natural numbers by making use of reflections...
factorization domains (UFD). Greatestcommondivisors exist in UFDs, and conversely, every integral domain in which greatestcommondivisors exist is an UFD. Every...
Lowest common factor may refer to the following mathematical terms: Greatestcommondivisor, also known as the greatestcommon factor Least common multiple...
greatestcommondivisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors...