Will reading "Elementary Number Theory" by David M. Burton be useful for Cryptography?
It's better to study cryptography and refer to number theory (and other fields) when it's needed. 

Nevertheless, Math is very important, if you want to be something better than 'press any key' cryptographer. Mostly, you will need Group Theory, Linear Algebra, Graph Theory, Automata Theory, Complexity, Number Theory, some bits of Functional/Mathematical analysis, a bit of Complex analysis, some knowledge about polynomials, and of course, Probability Theory.

Before obtaining a degree in Security I got Masters in Math and, looking back, it helped me a lot, compared to other students. And yes, your approach will be useful.