a. Prove that (a+b)^p is congruent to a^p + b^p (mod p) for any integers a, b, provided p is prime.

b. More generally, prove that if p is prime and q= p^n, where n is a natural number, then (a+b)^q is congruent to a^q + b^q (mod p) for any integers a,b. (Hint: Use Induction.)

Thanks in advance!