Question:
How many integers $n$ satisfy $0<n<60$ and $4n\equiv 2\pmod 6?$

Answer:
The residue of $4n\pmod 6$ is determined by the residue of $n\pmod 6.$ 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, $4n\equiv 2\pmod 6$ is true when $n\equiv 2$ or $n\equiv 5\pmod 6.$ 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 $$2, 5, 8, 11, 14, 17, \ldots, 56, 59.$$There are $\boxed{20}$ integers in this list.