2. A stair-shaped brick is composed of 12 unit cubes, with a width of 2 and 3 steps (as shown in Fig. 1). Find all positive integers $n$, such that the brick can be used to form a cube with edge length $n$.

保留源文本的换行和格式，翻译结果如下：

2. A stair-shaped brick is composed of 12 unit cubes, with a width of 2 and 3 steps (as shown in Fig. 1). Find all positive integers $n$, such that the brick can be used to form a cube with edge length $n$.

---

Note: The translation preserves the original text's line breaks and formatting.
Because the volume of a single brick is 12. Let a cube with edge length $n$ be composed of $m$ bricks, from $12 m=n^{3}$ we know that $n$ is a multiple of 6. Let $n=6 l$, where $l$ is a positive integer. On the other hand, two bricks can form a $2 \times 3 \times 4$ rectangular prism, and several such prisms can form a cube with edge length 12, so a cube with edge length a multiple of 12 can also be formed by such prisms. Below we prove that this condition is necessary, i.e., we prove that $l$ is even. Satisfying $x, y, z:=0$, and one vertex at the origin $O(0,0,0)$, while the edges are parallel to the coordinate axes,

Based on the parity of the triplet $(i, j, k)$, each unit cube $[i, i+1] \times[j, j+1] \times[k, k+1]$ is colored one of eight colors. Each stair-shaped brick contains all eight colors, and six of these colors appear only once, while the remaining two colors each appear three times.

Select any one of these eight colors, let $p$ be the number of stair-shaped bricks where this color appears three times, in the cube formed by $m$ bricks, the total number of times this color appears is
$$
3 p+(m-p)=m+2 p \text {. }
$$

On the other hand, in each $6 \times 6 \times 6$ cube, these eight colors each appear 27 times, so in a cube with edge length $6 l$, the number of times these eight colors appear is equal. Therefore, each color appears exactly $\frac{12 m}{8}$ times. Hence, $m+2 p=\frac{12 m}{8}$, i.e., $m=4 p$. Thus, $18 l^{3}=4 p$, from which we know that $l$ is even.