Diophantus, a Greek mathematician who lived during the 4th century A.D., was one of the first people who attempted to find integral or rational solutions to a given system of equations. Often the system involves more unknowns than equations. We will consider a linear equation, , with two unknowns .
proof: The first part of the theorem follows from lemma 1.2.4. Let , be any solution, and let be any other solution. We want to show that and , where . Substitute into the equation: :
proof: Assume . Let . By the above theorem, there exist integers , such that . Since divides and divides , by the hypthothesis, it divides , by Lemma 1.2.4. Therefore .
The following corollary is an immediadiate consequence of the previous one.