Find least nonnegative residue... n=99^999999 m=23

Let n be an integer. Prove each congruence below.

a. n^21 is congruent to n mod 30

b. n^7 congruent to n mod 42

c. n^13 is congruent to n mod 2730

Let a E Z and let p be a prime number. Prove that p|a^p + (p-1)!a and p|(p-1)!a^p +a