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The number of steps to calculate the GCD of two natural numbers, a and b, may be denoted by T(a, b). [96] If g is the GCD of a and b, then a = mg and b = ng for two coprime numbers m and n. Then T(a, b) = T(m, n) as may be seen by dividing all the steps in the Euclidean algorithm by g. [97]
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.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that
The Frobenius number exists as long as the set of coin denominations is setwise coprime. There is an explicit formula for the Frobenius number when there are only two different coin denominations, and , where the greatest common divisor of these two numbers is 1: . If the number of coin denominations is three or more, no explicit formula is known.
The slope of the tangent line at some point A=(x, y) is s = (3x 2 + 5)/(2y) (mod n). Using s we can compute 2A. If the value of s is of the form a/b where b > 1 and gcd(a,b) = 1, we have to find the modular inverse of b. If it does not exist, gcd(n,b) is a non-trivial factor of n. First we compute 2P.
The algorithm attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n.The algorithm works in two phases: the data collection phase, where it collects information that may lead to a congruence of squares; and the data processing phase, where it puts all the data it has collected into a matrix and solves it to obtain a ...
The greatest common divisor of p and q is usually denoted "gcd(p, q)". The greatest common divisor is not unique: if d is a GCD of p and q , then the polynomial f is another GCD if and only if there is an invertible element u of F such that f = u d {\displaystyle f=ud} and d = u − 1 f . {\displaystyle d=u^{-1}f.}