greatest common divisor

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.

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

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

• maximal common divisor

• 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