12.140 Given a rectangular box that can be filled with unit cubes. If it is instead filled with as many cubes of volume 2 as possible, with each cube's edges parallel to the box's edges, then the box's volume is exactly filled to $40 \%$, try to find the volume of the box with this property $(\sqrt[3]{2}=1.2599 \cdots)$
[Solution] Let $Q$ be a rectangular box with the properties described in the problem, with its three sides being $a, b, c$ and $a \leqslant b \leqslant c$. Since the box can be filled with unit cubes, $a, b$ are natural numbers. The edge length of a cube with volume 2 is $\sqrt[3]{2}$, so $a \geqslant 2$. When placing cubes with volume 2 into the box, the maximum number that can be placed is
$$
\left[\frac{a}{\sqrt[3]{2}}\right]\left[\frac{b}{\sqrt[3]{2}}\right]\left[\frac{c}{\sqrt[3]{2}}\right]
$$

Since their volume is 40% of the box's volume, we have
$$
2\left[\frac{a}{\sqrt[3]{2}}\right]\left[\frac{b}{\sqrt[3]{2}}\right]\left[\frac{c}{\sqrt[3]{2}}\right]=0.4 a b c .
$$

For easier discussion, we rewrite (1) as
$$
\frac{a}{\left[\frac{a}{\sqrt[3]{2}}\right]} \cdot \frac{b}{\left[\frac{b}{\sqrt[3]{2}}\right]} \cdot \frac{c}{\left[\frac{c}{\sqrt[3]{2}}\right]}=5 .
$$

To find the natural number solutions $\{a, b, c\}$ for (2), consider the function
$$
g(n)=\frac{n}{\left[\frac{n}{\sqrt[3]{2}}\right]}, \quad n=2,3, \cdots .
$$

To discuss the properties of the function $g(n)$, we list the values as follows:
From the table, we can see that starting from $n=6$, we have $g(n) \leqslant \frac{3}{2}$, and when $n \geqslant 7$, $g(n) < 1.25$, so (1) is impossible. From (2) and the table, we know that the solutions to (5) are two sets: $b=5, c=6$ and $b=3, c=5$.
In summary, the volume of the rectangular box is $2 \times 3 \times 5=30$ or $2 \times 5 \times 6=60$.