What are the relations between these number theorems and ECC?
Only on a high level (because I won't do your homework for you)

principal rest and modulus create groups in particular F_p which are fields arising from the the ring of integer modulo a prime. You can look at elliptic curves over such a field (which is a discrete set of points) and define a geometric point addition which (surprisingly?) is a group again. If you find cyclic subgroup with a generator of high (prime) order it will be useful for cryptography for two reasons: 


The discrete log problem seems to be difficult
Related to 1.) factorizing the large generator into its primes is hard (which is related to the fact that computing the greatest common divisor) is also hard.


I hope you see the gist where this is going. I strongly recommend reading up the wikipedia articles of the topics you mention. But connecting the dots is obviously  up to you.