jmc

The residue of $4n(\mathrm{mod}6)$ is determined by the residue of $n(\mathrm{mod}6).$ We can build a table showing the possibilities: \begin{array}{r || c * 5 {| c}} n\pmod 6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 4n\pmod 6 & 0 & 4 & 2 & 0 & 4 & 2 \end{array}As the table shows, $4n\equiv 2(\mathrm{mod}6)$ is true when $n\equiv 2$ or $n\equiv 5(\mathrm{mod}6).$ Otherwise, it's false.

So, our problem is to count all $n$ between $0$ and $60$ that leave a remainder of $2$ or $5$ modulo $6.$ These integers are $$2,5,8,11,14,17,\dots ,56,59.$$There are $\boxed{20}$ integers in this list.