equivalence relation-showing that an operation is well-defined

Question

Define $f: \mathbb{Z}_n \to \mathbb{Z}_n$ as $f([a]) = [a^2]$. Show that $f$ is a well-defined function. I am confused as to how I could show this.

Answer 1

You need to show that if $[a]=[b]$ in $\Bbb Z_n$ (i.e., $a\equiv b\pmod n$, i.e. $n\,|\,b-a$), then $[a^2]=[b^2]$.

For this, show first that $[a]=[b]$ implies $[a^2]=[ab]$.