Greatest common divisor

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 $\gcd(x,y)$ . 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.

[Long_1972_33-1] Long (1972, p. 33)

[Pettofrezzo_1970_34-2] Pettofrezzo & Byrkit (1970, p. 34)

[3] Kelley, W. Michael (2004), The Complete Idiot's Guide to Algebra, Penguin, p. 142, ISBN 978-1-59257-161-1.

[4] Jones, Allyn (1999), Whole Numbers, Decimals, Percentages and Fractions Year 7, Pascal Press, p. 16, ISBN 978-1-86441-378-6.

[Hardy&Wright_1979_20-5] Cite error: The named reference Hardy&Wright 1979 20 was invoked but never defined (see the help page).

[6] 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).

[7] 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.

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