Once you know how to solve diophantine equations with a single variable, the next step in complexity is to consider equations with two variables. The simplest such equations are linear and take the form ax+by=c. Before we solve this equation generally, we need a preliminary result. We show that you can solve the equation ax+by=GCD(a,b) by performing the Euclidean algorithm, and then reverse-substituting to arrive at a single solution.
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Number Theory is the study of whole numbers - also called integers. Since math began with the study of shapes and whole numbers, this is one of the oldest subjects around. However, it's also one of the most challenging. Many problems in number theory are easy to understand, but require a lot of cleverness to solve. It also does not require any advanced math to learn! Number Theory is for everyone.