$1 \cdot 101$ There are a sufficient number of rectangular prisms with edge lengths of $2,3,5$. In a cube with an edge length of 90, these rectangular prisms are neatly arranged in the same direction, filling it up until it is full. Find the number of rectangular prisms that a diagonal of the cube passes through.
[Solution] Divide the three edges of the cube into $\frac{90}{5}, \frac{90}{3}, \frac{90}{2}$ equal parts, i.e., $18, 30, 45$ parts, respectively. Then the cube is divided into
$$
\frac{90^{3}}{2 \times 3 \times 5}=24300
$$

rectangular prisms of size $2 \times 3 \times 5$.
The number of these rectangular prisms that a diagonal of the cube passes through is
$$
\begin{array}{l}
\quad\left(\frac{90}{5}-1\right)+\left(\frac{90}{3}-1\right)+\left(\frac{90}{2}-1\right)-\left[\left(\frac{90}{5 \times 3}-1\right)+\left(\frac{90}{3 \times 2}-1\right)\right. \\
\left.+\left(\frac{90}{5 \times 2}-1\right)\right]+\left(\frac{90}{2 \times 3 \times 5}-1\right)=65 .
\end{array}
$$