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function gcd(a, b) if b = 0 return a else return gcd(b, a mod b) (As above, if negative inputs are allowed, or if the mod function may return negative values, the instruction return a must be replaced by return max(a, −a).) For illustration, the gcd(1071, 462) is calculated from the equivalent gcd(462, 1071 mod 462) = gcd(462, 147).
gcd(a, b) is closely related to the least common multiple lcm(a, b): we have gcd(a, b)⋅lcm(a, b) = | a⋅b |. This formula is often used to compute least common multiples: one first computes the GCD with Euclid's algorithm and then divides the product of the given numbers by their GCD. The following versions of distributivity hold true:
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.
These formulas may be applied repeatedly to get a simple expression for any cyclotomic polynomial () in terms of a cyclotomic polynomial of square free index: If q is the product of the prime divisors of n (its radical), then [5] = (/).
In number theory, the gcd-sum function, [1] also called Pillai's arithmetical function, [1] is defined for every by = = (,) or equivalently ...
Therefore, equalities like d = gcd(p, q) or gcd(p, q) = gcd(r, s) are common abuses of notation which should be read "d is a GCD of p and q" and "p and q have the same set of GCDs as r and s". In particular, gcd(p, q) = 1 means that the invertible constants are the only common divisors.
As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers a, b is a Gaussian integer d that is a common divisor of a and b, which has all common divisors of a and b as divisor. That is (where | denotes the divisibility relation), d | a and d | b, and; c | a and c | b implies c | d.
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The number of integers coprime with a positive integer n, between 1 and n, is given by Euler's totient function, also known as Euler's phi function, φ(n).