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The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer. The GCD of a and b is generally denoted gcd (a, b).
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 (numbers), the largest number that divides them both without a remainder.
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [ 1 ] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, below ...
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, [1][2] is an algorithm that computes the greatest common divisor (GCD) of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons ...
Find first set. In computer software and hardware, find first set (ffs) or find first one is a bit operation that, given an unsigned machine word, [ nb 1 ] designates the index or position of the least significant bit set to one in the word counting from the least significant bit position. A nearly equivalent operation is count trailing zeros ...
Lehmer's GCD algorithm. Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. It is mainly used for big integers that have a representation as a string of digits relative to some chosen numeral system base, say β = 1000 or β = 2 32.
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under ...
A greatest common divisor of p and q is a polynomial d that divides p and q, and such that every common divisor of p and q also divides d. Every pair of polynomials (not both zero) has a GCD if and only if F is a unique factorization domain. If F is a field and p and q are not both zero, a polynomial d is a greatest common divisor if and only ...