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In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix that may also be referred to as Smith's matrix. The study was initiated by H.J.S. Smith (1875). A new inspiration was begun from the paper of Bourque & Ligh (1992). This led to intensive investigations on singularity and divisibility of GCD type ...
The so-called totatives 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is {1, 5, 7, 11}. The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are: {13,17,19,23} {−11,−7 ...
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.
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.
Corollary [6] — Suppose is a GCD domain (e.g., a unique factorization domain) with the field of fractions . Then a non-constant polynomial f {\displaystyle f} in R [ x ] {\displaystyle R[x]} is irreducible if and only if it is irreducible in F [ x ] {\displaystyle F[x]} and the gcd of the coefficients of f {\displaystyle f} is 1.
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.
In other words, every GCD domain is a Schreier domain. For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that dm = xy, or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is an LCM of x and y, and vice versa.
Fourier transforms of the GCD [ edit ] We can also express the Fourier coefficients in the equation immediately above in terms of the Fourier transform of any function h at the input of gcd ( n , k ) {\displaystyle \operatorname {gcd} (n,k)} using the following result where c q ( n ) {\displaystyle c_{q}(n)} is a Ramanujan sum (cf. Fourier ...