My question is specifically regarding congruence and the use of cases with the definition. I am only using the below mentioned example to emphasis my question.
I have a definition of congruence in my notes but I think I did not copy it down correctly.
Thm: $\forall a \in \mathbb{Z}, n \in \mathbb{N}, \exists q \in \mathbb{Z}, r \in \{0,1,...,(n-1)\}\,\textrm{such that}\,a= nq+r...$ I think I'm missing something in my definition.
I'm just a bit confused why do we have these cases in the proof? Can someone please comment on the use of cases?
if $a \in \mathbb{Z}$, then $a^3 \equiv a\ (\textrm{mod}\ 3)$.
\begin{align} a & \equiv r\ (\textrm{mod}\ n)\\\\ a^3 &\equiv 0\ (\textrm{mod}\ 3)\\ a^3 &\equiv 1\ (\textrm{mod}\ 3)\\ a^3 &\equiv 2\ (\textrm{mod}\ 3) \end{align}