Copyright © Philip M. Parker, INSEAD. Terms of Use.

Greatest Common Divisor

Definition: Greatest Common Divisor

Greatest Common Divisor

Noun

1. The largest integer that divides without remainder into a set of integers.

Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved.
 


Specialty Definitions: Greatest Common Divisor

DomainDefinitions

Computing

Greatest common divisor (GCD) A function that returns the largest positive integer that both arguments are integer multiples of. See also Euclid's Algorithm. Compare: lowest common multiple. (1999-11-02). Source: The Free On-line Dictionary of Computing.

Math

(1) The greatest integer which is a divisor of given positive integers. For instance, GCD(30, 42) = 6. (2) An algorithm to find the same. (references)

Source: compiled by the editor from various references; see credits.

Top     

Specialty Definition: Greatest common divisor

(From Wikipedia, the free Encyclopedia)

In mathematics, the greatest common divisor (abbreviated GCD), or highest common factor (HCF) of two integers which are not both zero is the largest integer that divides both numbers.

The GCD of a and b is often written as gcd(a,b). For example, gcd(12,18) = 6, gcd(-4,14) = 2 and gcd(5,0) = 5. The GCD of 0 and 0 is usually defined to be 0. Two numbers are called coprime or relatively prime if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime.

The greatest common divisor is useful for writing fractions in lowest terms. Consider for instance

42/56 = 3/4
where we cancelled 14, the greatest common divisor of 42 and 56.

Calculating the GCD

While the GCD of two numbers can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, this is never done in practice, because it is too slow. A much more efficient method is the Euclidean algorithm. An extended version of this algorithm can also compute integers p and q such that ap + bq = gcd(a, b).

Properties

Every common divisor of a and b divides the GCD of a and b.

The greatest common divisor of a and b (not both zero) may be defined alternatively and equivalently as follows: it is the smallest positive integer d which can be written in the form ap+bq where p and q are integers.

If d is the GCD of a and b, and a divides the product bc, then a/d divides c.

If m is any integer, then gcd(ma,mb) = m gcd(a,b) and gcd(a+mb,b) = gcd(a,b). If m is a nonzero common divisor of a and b, then gcd(a/m,b/m) = gcd(a,b)/m.

The GCD of three numbers can be computed as gcd(a,b,c) = gcd(gcd(a,b),c) = gcd(a, gcd(b, c)).

The GCD of a and b is closely related to their least common multiple lcm(a, b): we have

gcd(a, b) × lcm(a, b) = ab.
Furthermore, the following versions of distributivity hold true:
gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c))
lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c)).

Geometrically, gcd(a,b) is the number of points with integral coordinates on the straight line joining the points (0,0) and (a,b), excluding (0,0).

GCD in commutative rings

The greatest common divisor can more generally be defined for elements of an arbitrary commutative ring.

If R is a commutative ring, and a and b are in R, then an element of c of R is called a common divisor of a and b if it divides both a and b (that is, if there are elements x and y in R such that cx = a and cy = b). If c is a common divisor of a and b, and every common divisor of a and b divides c, then c is called a greatest common divisor of a and b.

Note that the GCD of a and b need not be unique, but if R is an integral domain then any two GCDs of a and b must be associate elements. Also, a and b need not have a GCD at all unless R is a unique factorization domain. If R is a Euclidean domain then a form of the Euclidean algorithm can be used to compute the GCD.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Greatest common divisor."

Top     

Synonyms: Greatest Common Divisor

Synonyms: greatest common factor (n), highest common factor (n). (additional references)

Top     

Crosswords: Greatest Common Divisor

Specialty definitions using "greatest common divisor": binary GCD algorithm ♦ Euclid's Algorithm, extended Euclid's algorithm. (references)

Top     

Modern Translations: Greatest Common Divisor

Language Translations for "greatest common divisor"; alternative meanings/domain in parentheses.

Dutch

  

grootste gemene deler (ggd) (greatest common divisor (gcd), greatest common factor, highest common factor). (various references)

   

French

  

plus grand diviseur commun (pgdc) (greatest common divisor (gcd), greatest common factor). (various references)

   

German

  

groesster gemeinschaftlicher Teilergroesster gemeinschaftlicher Faktorgroesster gemeinsamer Faktor (greatest common divisor (gcd), greatest common factor, highest common factor). (various references)

   

Italian

  

massimo comun divisore (greatest common divisor (gcd), greatest common factor, highest common factor). (various references)

   

Japanese Kanji 

  

最大公約数 . (various references)

   

Japanese Katakana 

  

さい い"うやくすう. (various references)

   

Pig Latin

  

eatestgray ommoncay ivisorday

Source: compiled by the editor from various translation references.

Top     

Anagrams: Greatest Common Divisor

Scrabble® YAWL-Verified Anagrams

Words within the letters "a-c-d-e-e-g-i-i-m-m-n-o-o-o-r-r-s-s-t-t-v"

-5 letters: radiogoniometers.

Source: compiled by the editor from various references; see credits.

SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Top     

Alternative Orthography: Greatest Common Divisor


Hexadecimal (or equivalents, 770AD-1900s) (references)

47 72 65 61 74 65 73 74      43 6F 6D 6D 6F 6E      44 69 76 69 73 6F 72

Leonardo da Vinci (1452-1519; backwards) (references)

        

Binary Code (1918-1938, probably earlier) (references)

01000111 01110010 01100101 01100001 01110100 01100101 01110011 01110100 00100000 01000011 01101111 01101101 01101101 01101111 01101110 00100000 01000100 01101001 01110110 01101001 01110011 01101111 01110010

HTML Code (1990) (references)

&#71 &#114 &#101 &#97 &#116 &#101 &#115 &#116 &#32 &#67 &#111 &#109 &#109 &#111 &#110 &#32 &#68 &#105 &#118 &#105 &#115 &#111 &#114

ISO 10646 (1991-1993) (references)

0047 0072 0065 0061 0074 0065 0073 0074      0043 006F 006D 006D 006F 006E      0044 0069 0076 0069 0073 006F 0072

Encryption (beginner's substitution cypher): (references)

41847167867185862378179798180238758875858184

Top     

 

INDEX

1. Definition
2. Synonyms
3. Crosswords
4. Translations: Modern 		5. Anagrams
6. Orthography
7. Bibliography


  

Copyright © Philip M. Parker, INSEAD. Terms of Use.

 

 

 

 

Note to the press & webmasters - this dictionary can be linked, indexed, or referred to using the following non-English expressions:
woordeboek, fjalor, ‏معجم, ‏قاموس, diccionariu, речник, diccionari, diksyonario, diksinario, 字典, gérlyver, slovník, ordbog, woordenboek, shimiyuc p'anca, orðabók, orðbók, dictionnaire, wurdboek, wörterbuch, λεξικό, אוצר מילים, szótár, uqausiit tukingit, dizionario, 字引 , じい, じびき, じて", ディクショナリー , じり", じしょ, '"かい, ディクショナリ , 사 , dizionari, recnik, fockleyr, dikshonario, słownik, dicionário, dicţionar, dicziunari, словарь, lolomi fefiloi, foclair, abardair, faclair, briathrachan, pukuntau, leksikon, rečnik, vocabbulariu, diccionario, sí-chazamagâma, ordbok, lexikon, พจนานุกรม, sözlük, ansiklopedik sözlük, словник, довідник, có tính chất sách vở, geirlyfr, geiriadur, for dictionary;
definisie, qartësi, përcaktim, saktësi, ‏الوضوحية في الشيء, ‏حد, ‏تحديد, ‏تعريف, ‏التحديد, ‏الإيضاحية, яснота, сила, очертания, дефиниция, 定義 , 定义, definice, deskriptordefinition, definitie, määritelmä, définition, ορισμός, "'"ר", "'בל", meghatározás, definíció, definizione, 確定 , ディーゼル電気車 , デ'ドロ酢酸 , デフィニション , ディフィニション , ていぎ, かくてい, 의, geyrid, meenaghey, keeayllaght, baght, definishon, definição, definiţie, determinare, definire, определение, definicija, definición, definition, açıklama, belirleme, belirtme, kesinleştirme, tanım, tarif, seçiklik, tanımlama, чіткість, тлумачення, виразність, визначення, дефініція, ясність, чітка чутність, sự định rõ, sự định nghĩa, lời định nghĩa sự định, diffiniad, darnodiad, for definition;
vertaling, transferim, transmetim, ‏ترجمة من لغة أجنبية للغة الأم, ‏ترجمة, ‏إفتتان, транслация, огъване, превод, предаване, поддаване, тълкуване, превеждане, 翻译, překlad, oversættelse, translatie, taajuusmuutos, translaatio, traduction, oersetting, Übersetzung, μετάφραση, תור'מ ות, תר'ום, "עתק", "עתק, fordítás, traduzione, 翻訳 , へい"ういどう, やくしょ, やくしゅつ, "うどく, ほ"やく, トランスレーション , やくじゅつ, ほ"やくしょ, 번역, tradukshon, tradução, translaţie, tãlmãcire, traducere, сдвиг, трансляция, перемещение, перевод, tumačenje, traducción, översättning, tercüme, процес перекладу, переклад, пояснення, переміщення, sự dịch, sự biến th nh sự giải thích, trosiad, for translation;
Nederlands, Hollands, holandez, ‏هولندي, ‏اللغة الهولندية, холандски, немски език, холандски език, холандците, немски, Olandes, 菏蘭語 , 荷兰语, holandský, nizozemský, hollandsk, hollendskt, hollantilainen, néerlandais, Nederlânsk, holländisch, ολλανδικόσ, ολλανδόσ, holandisht, "ול "י, holland, hollenskur, Ollainnis, olandese, 네덜란", Belanda, Ollanish, Germaanish, Tatimana, nederlandsk, ulandes, hulandes, holandês, neerlandés, olandez, nemţesc, limba olandezã, german, голландский, holanđanin, u škripcu, holandski, holandés, bakratongo, holländsk, ชาวเนเธอร์แลน"์, เกี่ยวกับเนเธอร์แลน"์, รรยา, alman, eş, flemenkçe, holandaca, hollanda, karı, hollandalı, hollandalılara özgü olan, Hollandali, hollanda'ya ait, голландська мова, голландський, ngôn ngữ khó hiểu, "b xã", for Dutch;
Franse taal, Frans, franceze, francez, frëngjisht, frëng, frëngjishte, ‏فرنسي, ‏اللغة الفرنسية, ‏الشعب الفرنسي, gall, френски език, френски, Pranses, 法國 , 法文 , 法語 , 法语, francouzština, francouzský, franskur, franskt, ranskalainen, français, Frânsk, französisch, γάλλοσ, γαλλικόσ, γαλλική γλώσσα, γαλλίδα, צרפתי, צרפתית, francia, Fraincis, francese, フレコン化 , フランス" , 仏文 , 仏 , ふつぶ", フレンチ , フランセ , ふつ, "랑스, Perancis, Ny Frangee, Mooinjey ny Frank, frances, franses, francês, francezii, francezã, franţuzesc, franţuzeşte, французский, Falani, francuski jezik, francuski, francuzi, francés, sí-Fulentji, fransk, franska, fransızca, Fransiz, fransızca ile ilgili, fransız, fransa ile ilgili, французька мова, французький, Ffrengig, isiFulentshi, for French;
Duits, Duitser, Duitse taal, Germaan, gjerman, ‏ضرب من الرقص, ‏جرماني, ‏المانية, ‏الماني, ‏اللغة الألمانية, роден, германски, немски език, немски, немец, готически, германец, 德語 , 德语, 德文 , 德國 , nìmecký, nìmec, tysker, Duitse, týskur, týskt, týskari, saksalainen, Allemand, Dútsk, Deutsche, Deutsch, "ερμανός, gjermanisht, 'רמ י, 'רמ ית, német, þjóðverji, þýskur, GearmÚnach, GearmÚinis, tedesco, ジプシー音楽 , ジャーマン , 독일, todesch, Germaanagh, Garmane, Germaanish, Carmane, aleman, Niemiec, niemiecki, alemão, alemand, neamţ, немецкий, Siamani, germanski, alemán, Tudesku, Doysri, mjeremani, mdachi, sí-Jalimáne, tysk, เยอรมัน, าษาเยอรมัน, Alman, німкеня, німецький, німець, $sisters german$ chị em ruột, $cousin german$ anh chị em con chú bác ruột, sister, Almaenwr, isiJalimane, iliJalimane, iJalimane, for German;
Italianer, Italiaans, Italiaan, ‏شخص إيطالي, ‏اللغة الإيطالية, ‏الإيطالي, ‏إيطالي, Italianu, италиански език, италиански, италианец, Italyano, 意大利 , 意大利語 , 意大利语, italština, italský, ital, italiener, italienskt, italialainen, Italien, Italjaansk, italienisch, Ιταλός, italisht, איטלקי, איטלקית, olasz, Ítali, IodÚilis, italiano, 이탈리아, Iddaalish, Włoch, italianã, italienesc, italieneşte, italian, итальянский язык, итальянский, итальянец, Italia, italijanski, italijanski jezik, italijan, sí-Taliyáne, italienare, italiensk, italienska, เกี่ยวกับอิตาลี, ชาวอิตาลี, าษาอิตาลี, italyanca, italyan, італі"ць, італійська мова, італійський, італійка, for Italian;