2.3.

Since $x$ and $y$ are both natural numbers, and $5.13 \times 5 > 24$, therefore, $y \in \{4,3,2,1,0\}$. At this point, $[5.13 y] = 5 y$.
Similarly, $[2.018 x] = 2 x$.
Thus, the original equation becomes $2 x + 5 y = 24$.
After calculation, $(x, y) = (2,4), (7,2), (12,0)$.
2. If $x, y$ are both natural numbers, then the number of solutions $(x, y)$ for the equation
$$
[2.018 x]+[5.13 y]=24
$$

is $\qquad$ (where $[x]$ denotes the greatest integer not exceeding the real number $x$).