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• Euclidean_algorithm abstract "In mathematics, the Euclidean algorithm, or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.The GCD of two positive integers is the largest integer that divides both of them without leaving a remainder (the GCD of two integers in general is defined in a more subtle way).In its simplest form, Euclid's algorithm starts with a pair of positive integers, and forms a new pair that consists of the smaller number and the difference between the larger and smaller numbers. The process repeats until the numbers in the pair are equal. That number then is the greatest common divisor of the original pair of integers.The main principle is that the GCD does not change if the smaller number is subtracted from the larger number. For example, the GCD of 252 and 105 is exactly the GCD of 147 (= 252 − 105) and 105. Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers, so this repetition will necessarily stop sooner or later — when the numbers are equal (if the process is attempted once more, one of the numbers will become 0).The earliest surviving description of the Euclidean algorithm is in Euclid's Elements (c. 300 BC), making it one of the oldest numerical algorithms still in common use. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable. This led to modern abstract algebraic notions, such as Euclidean domains. The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.The algorithm has many theoretical and practical applications. It may be used to generate almost all the most important traditional musical rhythms used in different cultures throughout the world. It is a key element of the RSA algorithm, a public-key encryption method widely used in electronic commerce. It is used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences (Chinese remainder theorem) or multiplicative inverses of a finite field. It can also be used to construct continued fractions, in the Sturm chain method for finding real roots of a polynomial, and in several modern integer factorization algorithms. Finally, it is a basic tool for proving theorems in modern number theory, such as Lagrange's four-square theorem and the fundamental theorem of arithmetic (unique factorization).If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits (in base 10) of the smaller integer. This was proved by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Methods for improving the algorithm's efficiency were developed in the 20th century.By reversing the steps in the Euclidean algorithm, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e.g., the GCD of 252 and 105 is 21, and 21 = [5 × 105] + [(−2) × 252]. This important property is known as Bézout's identity.".
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• Euclidean_algorithm title "Euclid's algorithm".
• Euclidean_algorithm title "Euclidean Algorithm".
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• Euclidean_algorithm subject Category:Articles_containing_proofs.
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• Euclidean_algorithm comment "In mathematics, the Euclidean algorithm, or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF).".
• Euclidean_algorithm label "Algorithme d'Euclide".
• Euclidean_algorithm label "Algoritme van Euclides".
• Euclidean_algorithm label "Algoritmo de Euclides".
• Euclidean_algorithm label "Algoritmo de Euclides".
• Euclidean_algorithm label "Algoritmo di Euclide".
• Euclidean_algorithm label "Algorytm Euklidesa".
• Euclidean_algorithm label "Euclidean algorithm".
• Euclidean_algorithm label "Euklidischer Algorithmus".
• Euclidean_algorithm label "Алгоритм Евклида".
• Euclidean_algorithm label "خوارزمية أقليدس".
• Euclidean_algorithm label "ユークリッドの互除法".
• Euclidean_algorithm label "輾轉相除法".
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• Euclidean_algorithm depiction Euclid's_algorithm_Book_VII_Proposition_2_3.png.
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