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In mathematics, the Euclidean algorithm, [note 1] 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).
Lehmer's algorithm is based on the observation that the initial quotients produced by Euclid's algorithm can be determined based on only the first few digits; this is useful for numbers that are larger than a computer word. In essence, one extracts initial digits, typically forming one or two computer words, and runs Euclid's algorithms on ...
"Generating African rhythms using the euclidean algorithm". Archived from the original on 2013-11-14. Benjamin Wardhaugh (1 September 2006). "Music and Euclid's algorithm". Links to videos about and a Flash app for experimenting with Euclidean rhythms; Euclidean rhythm demo — interactive browser-based tool for experimenting with Euclidean rhythms
A second difference lies in the bound on the size of the Bézout coefficients provided by the extended Euclidean algorithm, which is more accurate in the polynomial case, leading to the following theorem. If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials (s, t) such that
Euclidean algorithm; Coprime; Euclid's lemma; Bézout's identity, Bézout's lemma; Extended Euclidean algorithm; Table of divisors; Prime number, prime power. Bonse's inequality; Prime factor. Table of prime factors; Formula for primes; Factorization. RSA number; Fundamental theorem of arithmetic; Square-free. Square-free integer; Square-free ...
The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.
An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use the Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity.
Archived from the original (PDF) on 5 July 2015; For a comparison of the complexity of the Aryabhata algorithm with the complexities of Euclidean algorithm, Chinese remainder theorem and Garner's algorithm: T. R. N. Rao and Chung-Huang Yang (2006). "Aryabhata Remainder Theorem: Relevance to Public Key Crypto-systems" (PDF).