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Euclidean algorithm

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is an example of an…

History, Applications & Standards

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Historical development

Mathematical applications

Algorithmic efficiency

Generalizations

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Euclidean algorithm

Nodes193
Edges192
Triples158
Avg. degree1.99
Density0.010363
Components1

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Euclidean algorithm

Top relations

related to Historical development · 23
Euclidean algorithm → Alexandria, Aristotle, BC, Book, Book VII, But, Claude Brezinski, Cnidus, Elements, Euclid, Euclid's Elements, Eudoxus, In, In Book, It, Pappus, Propositions, Pythagoras, The, The Euclidean
related to background · 11
Euclidean algorithm → For, GCD, GCF, GCM, HCD, HCF, If, Synonyms, The, The Euclidean, This
related to Noncommutative rings · 11
Euclidean algorithm → Choosing, Euclidean, Here, Hurwitz, Let, Similarly, Since, The, The Euclidean, They, This
related to External links · 10
Euclidean algorithm → Archived, Demonstrations, Eric, Euclid's, Euclid's Algorithm, MathPagesEuclid's Game, MathWorld, PlanetMath, The Euclidean Algorithm, Wayback Machine
related to Gaussian integers · 10
Euclidean algorithm → As, By, Euclidean, Fermat's, Gaussian, In, Pythagorean, The Euclidean, The Gaussian, This
related to Rational and real numbers · 10
Euclidean algorithm → Book, Elements, Euclid, Euclid's, Euclidean, First, If, Second, The, This
related to Factorization algorithms · 9
Euclidean algorithm → Calculating, Continued, Dixon's, Euclid's, GCD, Lenstra, Pollard's, Shor's, The Euclidean
related to Number of steps · 8
Euclidean algorithm → By, Euclidean, GCD, GCDs, If, The, Then, Therefore
related to Stern–Brocot tree · 7
Euclidean algorithm → Brocot, Euclidean, For, Stern, The, The Euclidean, This
related to Visualization · 7
Euclidean algorithm → Assume, For, GCD, The, The Euclidean, This, We

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Important terminology

algorithm euclidean integers numbers number two gcd remainder common euclid's divisor steps integer step used since greatest may one factorization

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Lagrange's four-square theoreminstance ofit can be used as a basic tool for proving theorems in number theory0.80text
the uniqueness of prime factorizations.The original algorithm was described only for natural numbersinstance ofit can be used as a basic tool for proving theorems in number theory0.80text
geometric lengthsinstance ofit can be used as a basic tool for proving theorems in number theory0.80text
Euclidean domainsinstance ofThis led to modern abstract algebraic notions0.80text
an ideal in the ring of integersinstance ofalso used for concepts0.80text
which is closely related to GCD.If gcdinstance ofalso used for concepts0.80text
the set of Hurwitz quaternionsinstance ofprovided that the generalized Riemann hypothesis holds.Noncommutative ringsThe Euclidean algorithm may be applied to some noncommutative rings0.80text
the set of Hurwitz quaternionsinstance ofNoncommutative ringsThe Euclidean algorithm may be applied to some noncommutative rings0.80text
Euclidean algorithmrelated to backgroundThe Euclidean0.60section
Euclidean algorithmrelated to backgroundGCD0.60section
Euclidean algorithmrelated to backgroundThe0.60section
Euclidean algorithmrelated to backgroundSynonyms0.60section

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