I am wondering how to describe the distinct classes of $[2]$ and $[3]$.
For $[2]$ I got $[2] = \{5k + 3: k\in \mathbb{Z}\}$.
For $[3]$ I got $[3] = \{5k + 2: k\in \mathbb{Z}\}$.
But apparently these are switched around. How come? This is from my exam that I recently wrote and I really thought I had it. Any help?
PS: Forgive me for the poor formatting. I am still trying to learn how to do it as I am relatively new to the site.
$$0^3 \equiv 0 \pmod{5}$$ $$1^3 \equiv 1 \pmod{5}$$ $$2^3 \equiv 3 \pmod{5}$$ $$3^3 \equiv 2 \pmod{5}$$ $$4^3 \equiv 4 \pmod{5}$$
\begin{align}[2]&=\{n:2^3\equiv n^3 \pmod{5}\}\\&=\{n:3 \equiv n^3 \pmod{5}\}\\ &=\{n: n \equiv 2 \pmod{5}\} \end{align}