Strayer Elementary Number Theory 1. 3. 36b.

 

Disclaimer: This problem put me over a barrel for awhile because I couldn't think of the 'simple' solution the author was probably looking for. It's just that from several semesters of algebra, set theory, and analysis, the minute I hear the term "infinity," my mind immediately switches to sets, functions, and cardinalities. I just can't help it. So, in a nutshell, this proof feels too "involved" because of all the functions and stuff. But it works.

I couldn't think of a proof just using simple stuff covered so far in this book like divisibility and the greatest common divisor. As much as I hate to do it, I eventually broke down and googled the problem for a hint. Turns out I didn't find this problem, but I found solutions to similar ones. So, using the strategies I found there, I developed a simpler solution probably more in line to what the author was looking for. Here it is: