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In mathematics, the **greatest common divisor** (**GCD**), also known as **greatest common factor (GCF)**, 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..