2. When $m$ takes all real values from 0 to 5, the number of integer $n$ that satisfies $3 n=m(3 m-8)$ is $\qquad$ .
2. 13 .

Let $n$ be a quadratic function of $m$, i.e.,
$$
n=m^{2}-\frac{8}{3} m \text {, }
$$

then
$$
n=\left(m-\frac{4}{3}\right)^{2}-\frac{16}{9} \text {. }
$$

From the graph, when $0 \leqslant m \leqslant 5$,
$$
\begin{array}{l}
-\frac{16}{9} \leqslant n \leqslant \frac{35}{3} . \\
\therefore n=-1,0,1,2,3,4,5,6,7,8,9,10,11 .
\end{array}
$$