So far, we have mostly investigated topics that will seem familiar even to the high school student; for instance, the gcd shows up in adding fractions with unequal denominators.

What makes number theory so interesting is that even a slight change in the questions we ask, or the way in which we approach them, can yield completely unexpected insights.

In this section, we will begin this process by going from the simple questions we started with into more subtle ones, largely motivated by a surprising connection with geometry.