Equivalence classes

Classes.
- $\overline 0=\{5k:k\in\mathbb{Z}\}$
- $\overline 1=\{5k+1:k\in\mathbb{Z}\}$
- $\overline 2=\{5k+2:k\in\mathbb{Z}\}$
- $\overline 3=\{5k+3:k\in\mathbb{Z}\}$
- $\overline 4=\{5k+4:k\in\mathbb{Z}\}$

Quotient. $\mathbb{Z}/5\mathbb{Z}=\{\overline 0,\overline 1,\overline 2,\overline 3,\overline 4\}$, cardinality $5$.

Conclusion. Five equivalence classes, partitioning $\mathbb{Z}$.