Let $p\geq 2$ be a natural number. Define $\equiv_p$ on $\mathbb{Z}$ by declaring $x\equiv_p$ if and only if there exists $k\in\mathbb{Z}$ such that $x-y=pk.$ For equivalence classes $[a]$ and $[b]$ define their product to be: $$[a].[b]=[a.b]$$ Prove that this is well-defined. That is, show that if $[a]=[a']$ and $[b]=[b']$, then $[a].[b]=[a'].[b']$
So I know that $a-a'=pn$ and $b-b'=pm$ and that I think I am trying to show that $[a.b]=[a'.b']$, but I am not sure how to reach that. I would appreciate the help.