I'm having difficulty with solving this problem, hope somebody could help.

Suppose (a,p)=1 where a is an integer and p is prime.

1) Show A={$\displaystyle m, 2m, 3m,..., (p-1)m$} is is reduced residue system mod p.

2) Let $\displaystyle C_k=1^k +2^k+...(p-1)^k$. Use part 1) to show that

$\displaystyle m^kC_k \equiv C_k mod p$

3) Let the number m be the primitive root g mod p. Prove that $\displaystyle C_k\equiv 0 mod p$ if p-1 doesn't divide k.

For the first part, can we pick a reduced residue system mod p, {1,2,3,...,p-1} then try to show that A={m,2m,3m,...,(p-1)m} again a r.r.s mod p. We know that {km} where k=1,2,...,p-1 are relatively prime to p. Then it suffices to show that no two elements from A are congruent mod p. Am I on the right track on this? I really have no idea on the last two questions.