BIGtheme.net http://bigtheme.net/ecommerce/opencart OpenCart Templates
Friday , July 28 2017
Home / Elementary Number Theory / If a and b are given integers not both zero, then the set T = {ax + by| x, y are integers} precisely the set of all multiplies of d = gcd(a, b)

If a and b are given integers not both zero, then the set T = {ax + by| x, y are integers} precisely the set of all multiplies of d = gcd(a, b)

Proof: Since d\a and d\b, we know that d\(ax + by) for all integers x, y. Thus every member of T is a multiple of d. On the other hand, d may be written as d = ax0 + by0 for suitable integers x0 and y0. So that any multiple nd of d is of the form nd = n(ax0 +by0) = a(nx0) + b(ny0)

Hence, nd is a linear combination of a and d, and by definition lies in T.    (Proved)

Check Also

Application of Euclidian’s algorithm in Diophantine equation

Problem-1: Which of the following Diophantine equations cannot be solved –   a) 6x + ...

Leave a Reply

Your email address will not be published. Required fields are marked *