Prove that the set of integers modulo m is precisely {[0], [1],...,[m-1]}

Question

I need to prove that each of the congruence classes listed above are different, and that any other congruence class [a](for any integer a) has to be equal to one of the congruence classes listed.

I know in order to prove the second part I must show that a is a subset of a congruence class and that congruence class is a subset of a to show they're equal but I'm not sure how to go about the first part.

Answer 1

The congruence class of an integer is the congruence class of its remainder when divided by $m$. Two remainders can't be congruent, since they're their own remainders. What is the list of the possibles remainders?