How to prove that a function on a quotient set $\mathbb{Z}/n\mathbb{Z}$ is well defined?

Question

I am asked to prove that a function under a quotient set is well-defined. I know that for a function to be well-defined, two mappings or images found in the co-domain/range can't be mapped from the same argument in the domain. But I don't know how to prove it under this multiplicative group: $f: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, $[a] \mapsto [a^2]$. Thank you

Answer 1

Hint:

If you have $[a]=[b]$ then you need to prove that $[a^2]=[b^2]$ too.

Remember that $[a]=[b]$ implies that $a-b$ is a multiple of the modulus $n$, that is, $a-b=kn$ for some integer $k$.