proof with multiplicative inverses mod n

Question

I am trying to write a proof for-

show that if [b] and [c] are both multiplicative inverses of [a] in Zn, then b congruent c (mod n)

I don't know a lot about multiplicative inverse proofs and any help will be appreciated.

Answer 1

Multiplication is commutative and associative. $$[b] = [b][1] = [b]([c][a]) = [c]([b][a]) = [c][1] = [c]$$