This is the first exercise of section 3.3 from A Book of Set Theory by Charles C. Pinter.
This is my attempt to solve it, please tell me where I'm doing wrong:
To prove that $A_n$ is a partition of $\mathbb{Z}$, we must prove that:
P1: $$If\:\exists x \in A_i \cap A_j, then \: A_i=A_j$$ P2: $$If \:x \in A, then\: x \in A_i \:for\: some\: i \in I$$
Suppose $A_i \cap A_j$ is a non-empty set. We must prove that $A_i = A_j$
Let $y \in A_i$. Then by definition of $A_n$, there exists some $k$ such that $y = i + 5k$. We should prove that $y \in A_j$. Add and substract 5 from the right hand side: $$y = i + 5k + 5 -5$$ $$y = i + 5 + 5(k - 1)$$
Now let $j = i + 1$. Then $y \in A_j$ for $j = i + 5$ and $q = k + 1$.
Proof of P2: Let $n \in \mathbb{Z}$. We must prove that $n \in A_i$ for some $i \in I$. But this can be easily shown to be true by the quotient-reminder theorem.