I mean instead of partitioning $\mathbb Z$ to $n$ different sets named residue classes and defining arithmetic operations on them, we can just define
$$\mathbb Z_n = \lbrace 0, 1, 2, ..., n-1 \rbrace$$
As a ring (a field if $n$ is prime) closed under binary relations modular addition and multiplication defined as follow,
$$X, Y \in \mathbb Z_n$$ $$(X + Y) = Qn + R$$ $$X +_n Y = R$$ ~ $$(X \cdot Y) = Q'n + R'$$ $$X \cdot_n Y = R'$$
Then other operations like division (if $\mathbb Z_n$ is a field), exponentiation etc. can be defined over all other elements. Why is this simply not sufficient and we need to make sure that any element of the $\mathbb Z_n$ have to represent all other elements of $\mathbb Z$ which yields the same remainder as itself when divided by $n$?
In other words why we need residue classes? What makes them more useful than the simple definition i denote above.