Will reading "Elementary Number Theory" by David M. Burton be useful for Cryptography? I am a Computer Science student and interested to learn Cryptography to be a researcher in this field. I heard that Number Theory is useful for Cryptography since there are some Crypto systems based on it. I really like Number Theory and have started reading Elementary Number Theory by Burton. So far I have been reading the theorems and trying to prove them myself. Will my approach 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.