Hello, i want to share some problems that i found in the internet, but i cannot verify the solutions and others i dont know how to solve them,

here:

a) If N is a odd natural number, show that a+b divides (a^n)+(b^n)

b) There exist a natural number N such that 1955 divides n^2+n+1 ?

c) Find the remainder when 4444^4444 is divided by 9.

d) p & q are different primes, show that

1) p^q + q^p i is congruent to p+q (mod pq)

2) (p^q + q^p)/(pq) is even, if p & q is not equal to 2.

a step by step solution will be great, please.. thank you!