greatest common divisor

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greatest common divisor

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Noun

greatest common divisor (plural greatest common divisors)

1. (arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring).

   The greatest common divisor of 66, 30 and 18 is 6.

   • 1974, John M. Peterson, Basic Concepts of Elementary Mathematics, Prindle, Weber & Schmidt, page 148,

     Euclid's algorithm is a process for finding the greatest common divisor of any two whole numbers.

   • 2006, W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Rings, Hexis, page 53:

     Suppose ${\displaystyle (a_{0},a_{1},\dots ,a_{n})\in \mathbb {Z} }$ we say ${\displaystyle (a_{0},a_{1},\dots ,a_{n})}$ is primitive if the greatest common divisor of ${\displaystyle a_{0},a_{1},\dots ,a_{n}}$ is 1.

   • 2011, Zhonggang Zeng, “The Numerical Greatest Common Divisor of Univariate Polynomials”, in Leonid Gurvits, Philippe Pébay, J. Maurice Rojas, David Thompson, editors, Randomization, Relaxation, and Complexity in Polynomial Equation Solving, American Mathematical Society, page 187:

     This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. […] As one of the fundamental algebraic problems with a long history, finding the greatest common divisor (GCD) of univariate polynomials is an indispensable component of many algebraic computations besides being an important problem in its own right.

Synonyms

• (largest number that divides each of a set of positive integers): gcd (abbreviation), greatest common factor, highest common factor

Antonyms

• (antonym(s) of “largest number that divides each of a set of positive integers”): least common multiple

Related terms

• maximal common divisor

Translations

largest positive integer or polynomial

Chinese: Mandarin: 最大公因子 (zh) (zuìdà gōngyīnzǐ), 最大公約數／最大公约数 (zh) (zuìdà gōngyuēshù) Czech: největší společný dělitel m Danish: største fælles divisor Dutch: grootste gemene deler Estonian: suurim ühiskordne Finnish: suurin yhteinen tekijä (fi) French: plus grand commun dénominateur m, plus grand commun diviseur (fr) m Georgian: უდიდესი საერთო გამყოფი (udidesi saerto gamq̇opi) German: größter gemeinsamer Teiler m Greek: μέγιστος κοινός διαιρέτης m (mégistos koinós diairétis) Hungarian: legnagyobb közös osztó (hu) Icelandic: stærsti samdeilir m, stærsti sameiginlegi deilir m Indonesian: faktor persekutuan terbesar Japanese: 最大公約数 (ja) (saidai kōyakusū) (integer), 最大公約多項式 (saidai kōyaku takōshiki) (polynomial) Korean: 최대공약수 (choedaegong'yaksu) Malay: faktor sepunya terbesar Malayalam: ഉത്തമ സാധാരണ ഘടകം (uttama sādhāraṇa ghaṭakaṁ) Norwegian: største felles faktor, største felles nevner Polish: największy wspólny dzielnik m Portuguese: máximo divisor comum (pt) m Romanian: cel mai mare divizor comun, c.m.m.d.c Russian: наибо́льший о́бщий дели́тель (ru) m (naibólʹšij óbščij delítelʹ) Spanish: máximo común divisor m Swedish: största gemensamma delare (sv) Turkish: en büyük ortak bölen, ortak bölenlerin en büyüğü Ukrainian: найбі́льший спі́льний дільни́к m (najbílʹšyj spílʹnyj dilʹnýk) Vietnamese: ước chung lớn nhất, ước số chung lớn nhất

Further reading

• Polynomial greatest common divisor on Wikipedia.Wikipedia
• Maximal common divisor on Wikipedia.Wikipedia
• Least common multiple on Wikipedia.Wikipedia
• Euclidean algorithm on Wikipedia.Wikipedia
• Binary GCD algorithm on Wikipedia.Wikipedia
• Bézout domain on Wikipedia.Wikipedia