greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers is the largest positive integer that divides the numbers without a remainder.

From the relationships (11) and (12) it follows that the polynomial cm(z) is the greatest common divisor of all elements of matrix C(z).

So if g is the greatest common divisor of the [[alpha].sub.i] (which can be computed in polynomial time), and [alpha]/g = [[[alpha].sub.1]/g, [[alpha].sub.2]/g, ..., [[alpha].sub.N+1]/g] the formula E([alpha])(gt) = E([alpha]/g)(t) holds, and we may assume that the numbers [[alpha].sub.i] span Z without changing the complexity of the problem.

We use the definition as in [6]; for any arbitrary n x n matrix A, form the characteristic matrix x[I.sub.n] - A and let [d.sub.j](x) denote the greatest common divisor (gcd) of all minors of order j of x[I.sub.n] - A, j = 1,2, ..., n.

We say that Y is an indecomposable curve of canonical type if [K.sub.X] x [Y.sub.i] = Y x [Y.sub.i] = 0 for every i, SuppY is connected, and the greatest common divisor of integers [n.sub.1,] ..., [n.sub.k] is equal to one.

Figure 4: The first seven members of the aliquot sequence of 30: sum of proper integer proper divisors divisors 30 1, 2, 3, 5, 6, 10, 15 42 42 1, 2, 3, 6, 7, 14, 21 54 54 1, 2, 3, 6, 9, 18, 27 66 66 1, 2, 3, 6, 11, 22, 33 78 78 1, 2, 3, 6, 13, 26, 39 90 90 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 144 45 144 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 259 24, 36, 48, 72 What is striking about the sequence {30, 42, 54, 66, 78, 90, 144} is that the Greatest Common Divisor (CGD) for its 7 members is 6, which is a 'perfect number' because it equals the sum of its proper divisors (1+2+3=6).

In Section 3, we discuss the concept of the greatest common divisors in the semimodules and show that for any Euclidean semimodule A in which every cyclic subsemimodule is subtractive, then every nonempty finite subset of A has a greatest common divisor.

Theorem 1 If c = a + b and d = gcd(a, b) the orbit of the billiard ball (on the corresponding table) passes through a lattice point (x, y) on the boundary if and only if d|x and d|y (gcd(a, b) denotes the greatest common divisor of a and b)

They studied the necessary and sufficient condition under which from p-periodicity and q-periodicity we can derive gcd(p, q)-periodicity (recall that gcd is the greatest common divisor).

where C(s) is a left greatest common divisor of matrixes, and [??](s) and [??](s) are mutually distinct.