Editing Greatest common divisor (section)

{{Use American English|date = March 2019}}
{{Short description|Largest integer that divides given integers}}
In [[mathematics]], the '''greatest common divisor''' ('''GCD'''), also known as '''greatest common factor (GCF)''', of two or more [[integer]]s, which are not all zero, is the largest positive integer that [[divides]] each of the integers. For two integers {{math|''x''}}, {{math|''y''}}, the greatest common divisor of {{math|''x''}} and {{math|''y''}} is denoted <math>\gcd (x,y)</math>. For example, the GCD of 8 and 12 is 4, that is, {{math|1=gcd(8, 12) = 4}}.<ref name="Long 1972 33">{{harvtxt|Long|1972|p=33}}</ref><ref name="Pettofrezzo 1970 34">{{harvtxt|Pettofrezzo|Byrkit|1970|p=34}}</ref>

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.<ref>
{{cite book
 | last = Kelley | first = W. Michael
 | isbn = 978-1-59257-161-1
 | page = 142
 | publisher = Penguin
 | title = The Complete Idiot's Guide to Algebra
 | url = https://books.google.com/books?id=K1hCltk-2RwC&pg=PA142
 | year = 2004}}.</ref><ref>{{cite book
 | last = Jones | first = Allyn
 | isbn = 978-1-86441-378-6
 | page = 16
 | publisher = Pascal Press
 | title = Whole Numbers, Decimals, Percentages and Fractions Year 7
 | url = https://books.google.com/books?id=l-ItSuk-zngC&pg=PA16
 | year = 1999
}}.</ref><ref name="Hardy&Wright 1979 20" /><ref>Some authors treat '''{{vanchor|greatest common denominator}}''' as synonymous with ''greatest common divisor''. This contradicts the common meaning of the words that are used, as  ''[[denominator]]'' refers to [[fraction (mathematics)|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]]).</ref>  Historically, other names for the same concept have included '''greatest common measure'''.<ref>
{{cite book
 | last1 = Barlow | first1 = Peter | author1-link = Peter Barlow (mathematician)
 | last2 = Peacock | first2 = George | author2-link = George Peacock
 | last3 = Lardner | first3 = Dionysius | author3-link = Dionysius Lardner
 | last4 = Airy | first4 = Sir George Biddell | author4-link = George Biddell Airy
 | last5 = Hamilton | first5 = H. P. | author5-link = Henry Hamilton (priest)
 | last6 = Levy | first6 = A.
 | last7 = De Morgan | first7 = Augustus | author7-link = Augustus De Morgan
 | last8 = Mosley | first8 = Henry
 | page = 589
 | publisher = R. Griffin and Co.
 | title = Encyclopaedia of Pure Mathematics
 | url = https://books.google.com/books?id=3fIUAQAAMAAJ&pg=PA589
 | year = 1847
}}.</ref>

This notion can be extended to polynomials (see ''[[Polynomial greatest common divisor]]'') and other [[commutative ring]]s (see ''{{section link|#In commutative rings}}'' below).