So you want to show that there exist a number "b" such that, for any a, [a]+ [b]= [a].

Since [a]+ [b]= [a+ b], that means you want to show that there exist a number [a+ b]= [a] for any a. That is, of course, the same as saying that a+ b= a (mod n). What must b be equal to?

Similarly, you want to show that there exists a number "c" such that [ac]= [a]. And that is the same as saying that ac= a (modulo n). What must c be equal to?

To show that there exist additive inverses, you need to show that, for any a, there exist a number c such that [a+ c]= b (where b is the number you found in the first part). That is, you need to show that there exist a+ c= b (mod n).