Let [alpha] = [[[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.N+1]] be a list of integers with

greatest common divisor equal to 1, and let

If B is a nonempty subset of a semimodule A over a semiring R, then an element b of A is a

greatest common divisor of B if and only if the following conditions are satisfied:

This textbook for an undergraduate introductory course in number theory begins with the ancient Euclidean algorithm for finding the

greatest common divisor of two integers, and ends with the theory of elliptic curves and some other modern developments.

The purpose of this article is to examine one possible extension of

greatest common divisor (or highest common factor) from elementary number properties.

The assumption that the

greatest common divisor of {k [member of] N | [a.sub.k] = [not equal to] 0} is 1 means that [theta] = 0.Therefore z = [r.sub.0], and we conclude that r0 is a unique zero of g(t) having the smallest absolute value of all zeros of g(t).

Let (a, b) = d denotes the

Greatest Common Divisor of a and b, and a = d x [a.sub.1], b = d x [b.sub.1].

Since we will make connections between periodicity of trigonometric functions and the concepts of

greatest common divisor and the least common multiple that are defined only for integers, we need to provide extended definition of the terms "divisor", "multiple", "

greatest common divisor" and the "least common multiple" of rational numbers.

, [x.sub.t]] denote the

greatest common divisor and the least common multiple of any positive integers [x.sub.1], [x.sub.2], ...

Assume that (u, v) is the

greatest common divisor of u and v, then we have

To understand the statement of Dirichlet's theorem, recall that given two integers a and b, their

greatest common divisor is the largest integer which is a divisor of both a and b; we denote it gcd(a,b).

where (s(n),m) denotes the

greatest common divisor of s(n) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [epsilon] is any positive number.