Greatest Common Divisor

Synonyms

GCD; Greatest common factor

Related Concepts

Euclidean Algorithm; Least Common Multiple; Number Theory; Relatively Prime

Definition

The greatest common divisor (gcd) of a set of positive integers $\{{a}_{1},\ldots,{a}_{k}\}$ is the largest integer that divides every element of the set. This is denoted $\gcd ({a}_{1},\ldots,{a}_{k})$ or sometimes just $({a}_{1},\ldots,{a}_{k})$.

Background

Approximately in 300 BC, the Greek mathematician Euclid described a method for computing the greatest common divisor in Book 7 of Elements. Known as the Euclidean algorithm, it is based upon the fact that the greatest common divisor of two integers is preserved under subtraction.

Theory

An important property of the greatest common divisor is that it can always be written as an integer linear combination of the elements of the set. In other words, there exist integers ${x}_{1},\ldots,{x}_{k}$ such that ${\sum \nolimits }_{i=1}^{k}{a}_{i} \cdot {x}_{i} =\gcd ({a}_{1},\ldots,{a}_{k})$.