Euclidean algorithm
Euclid’s algorithm is an efficient method for computing the greatest common divisor of two integers. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements. The algorithm was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. It has many theoretical and practical applications, including being used to prove Lagrange’s four-square theorem.
About Euclidean algorithm in brief
Euclid’s algorithm is an efficient method for computing the greatest common divisor of two integers. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. No natural number than 1 divides both 6 and 35, since they have no prime factors in common, and there is no larger number for which there is a coprime. The algorithm was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. It was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. Additional methods for improving the algorithm’s efficiency were developed in the 20th century. The Euclidean algorithm is used for proving theorems in number theory such as Lagrange’s four-square theorem and the uniqueness of prime factorizations. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. If gcd = 1, then a and b are said to be coprIME. This property does not imply that a or b are themselves prime numbers. Let gcd = gcd, since they are both multiples of g, they can be written a=mg and b=mg.
The greatest common factor is the largest natural number that divides both a andb without leaving a remainder, and this is true for natural numbers and nmg as well. The most common factor of a and a is gmg, which is the most common measure of the two numbers, and can also be written gcd or, more simply, as, although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers. The great common factor can be expressed as a linear combination of two original numbers, each multiplied by an integer × 252). The fact that the GCD can always be expressed in this way is known as Bézout’s identity, and it is also known as the “Euclidean identity” It is used to solve Diophantine equations, find numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. For example, 6, 6 =7, 6 =7 and 35 =5 are all coprimes, but they are not prime numbers, since there are no other factors that can divide both m and n to make gmg greater than mmg. This is true since any other number that divides m and b must also divide both c and b also divide c and n, and therefore gmg must be greater than nmg. It has many theoretical and practical applications, including being used to prove Lagrange’s four- square theorem.
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This page is based on the article Euclidean algorithm published in Wikipedia (as of Nov. 16, 2020) and was automatically summarized using artificial intelligence.
