10. There are a sufficient number of rectangular prisms with edge lengths of $2, 3, 5$. In a cube with a side length of 90, these rectangular prisms are neatly arranged in the same direction, filling it up. The number of rectangular prisms that a diagonal of the cube passes through is
10. 65 Let the three dimensions of the cube be denoted as $x, y, z$, and divide them into 18, 30, and 45 equal parts, respectively. Then the cube is divided into 270 $2 \times 3 \times 5$ rectangular prisms. Using the principle of inclusion-exclusion, it is easy to find that the number of intersection points is $\left(\frac{90}{5}-1\right)+\left(\frac{90}{3}-1\right)+\left(\frac{90}{2}-1\right)-$ $\left[\left(\frac{90}{5 \times 3}-1\right)+\left(\frac{90}{3 \times 2}-1\right)+\left(\frac{90}{5 \times 2}-1\right)\right]+\left(\frac{90}{2 \times 3 \times 5}-1\right)=65$.