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The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c. [6] The greatest common divisor can be visualized as follows. [7] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly.
Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. Thus, the GCD is 2 2 × 3 = 12. 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.
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.
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 .
For example, if you had two types of coins valued at 6 cents and 14 cents, the GCD would equal 2, and there would be no way to combine any number of such coins to produce a sum which was an odd number; additionally, even numbers 2, 4, 8, 10, 16 and 22 (less than m=24) could not be formed, either.
A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
The above formulas lead to an efficient O(log a log b) [3] algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the gcd of two numbers. (This should not be surprising in light of rule 2.) Reduce the "numerator" modulo the "denominator" using rule 2. Extract any even "numerator" using rule 9.
A simple and sufficient test for the absence of a dependence is the greatest common divisor (GCD) test. It is based on the observation that if a loop carried dependency exists between X[a*i + b] and X[c*i + d] (where X is the array; a, b, c and d are integers, and i is the loop variable), then GCD (c, a) must divide (d – b).