Euclidean algorithm
Noun
Meaning
(historical) Any of certain algorithms first described in Euclid's Elements.
(arithmetic, number theory) Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method.
Source: en.wiktionary.orgExamples
A key advantage of the Euclidean algorithm is that it can find the GCD efficiently without having to compute the prime factors.
After each step k of the Euclidean algorithm, the norm of the remainder f(r k ) is smaller than the norm of the preceding remainder, f(r k−1 ).
Description Procedure The Euclidean algorithm proceeds in a series of steps such that the output of each step is used as an input for the next one.
History An early example of algorithm complexity analysis is the running time analysis of the Euclidean algorithm done by Gabriel Lamé in 1844.
If both a and b are nonzero, the extended Euclidean algorithm produces one of the two pairs such that and (equality may occur only if one of a and b is a multiple of the other).
In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the Euclidean algorithm ; Elements, Prop. VII.2) and the first known proof of the infinitude of primes (Elements, Prop. IX.20).