Although I can prove these algebraically via induction, how are the following proved using modular arithmetic (where "=" means "congruent to")?

1+2+...+(n-1)=0 mod n iff n is odd

1^2+2^2+...+(n-1)^2=0 mod n iff n=+/-1 mod 6

1^3+2^3+...+(n-1)^3=0 mod n iff n is not congruent to 2 mod 4