[Math] How to prove a set of numbers is a complete system of residues modulo of a number

Question

Definition:
A complete system of residues modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set.

Example: The division algorithm shows that the set of $0, 1, 2, …, m – 1$ is a complete system of residues modulo m.

What I did not understand is "congruent modulo m to exactly one integer of the set". Could anyone give me a counter example to this?

And here is the problem:

Prove that the set $0, 1, 3, 3^2, 3^3, …., 3^{15}$ is a complete system of residues modulo $17$.

I really have no idea how to start and what to prove :(. To be a complete system of residues modulo, what property does this set have to have? A hint would be greatly appreciated.

Thanks,

Answer 1

HINT: $\left(\frac{3}{17}\right) = -1$. Essentially you want to prove all the remainders are distinct $\bmod 17$.