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, 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.

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 are equal. That number then is the greatest common divisor of the original pair.

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 (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.

The GCD of two numbers is the largest number that divides both of them without leaving a remainder. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number. For if k, m, and n are integers, and k is a common factor of two integers A and B, then A = nk and B = mk implies AB = (nm)k, therefore k is also a common factor of the difference. That k may also represent the greatest common divisor is proven below. For example, 21 is the GCD of 252 and 105 (252 = 12 × 21; 105 = 5 × 21); since 252 − 105 = (12 − 5) × 21 = 147, the GCD of 147 and 105 is also 21.

Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers until one of them is zero. When that occurs, the GCD is the remaining nonzero number. 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., 21 = + . This important property is known as Bézout's identity.

Read more about Euclidean Algorithm:  Background — Greatest Common Divisor, Description, Historical Development, Algorithmic Efficiency, Other Number Systems, Generalizations To Other Mathematical Structures

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