Ana has a cube with a side length of $10 \mathrm{~cm}$. She cut the cube into smaller cubes with a side length of $1 \mathrm{~cm}$ and plays by forming other rectangular blocks with these smaller cubes, without any cubes left over. For example, she formed a block of $10 \times 20 \times 5$. In total, how many different blocks can she build with these smaller cubes, without any cubes left over?

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-028.jpg?height=254&width=662&top_left_y=787&top_left_x=731)
The volume of the cube is $10 \times 10 \times 10=1000 \mathrm{~cm}^{3}$. The volume $V$ of a block is the product of its three dimensions, height $(=a)$, width $(=l)$, and length $(=c)$.

To construct each block, Ana must use all the small blocks, so the volume of each block is

$$
V=\text { height } \times \text { width } \times \text { length }=a l c=1000 \mathrm{~cm}^{3}
$$

Thus, we need to know in how many ways we can write 1000 as the product of three positive integers $a, l$, and $c$. For this, we factorize 1000, obtaining $a l c=1000=2^{3} \times 5^{3}$.

Solution 1: We can find all these numbers by listing the dimensions $a, l$, and $c$ of the blocks. Without loss of generality, we can assume that $a \leq l \leq c$. Then $a^{3} \leq a l c \leq 1000$ and, therefore, $a \leq 10$. Thus, $a=1,2,4,5,8$ or 10. But if $a=8$, then $l c=125=5^{3}$ and, since $8 \leq l \leq c$, there is no way to obtain integer measures for $l$ and $c$. Therefore, $a$ can only be $1,2,4,5$ or 10. The table shows the 19 possibilities for these blocks.

| $a$ | $l$ | $c$ | $a$ | $l$ | $c$ |
| ---: | ---: | ---: | ---: | ---: | ---: |
|  | 1 | 1000 |  | 2 | 250 |
|  | 2 | 500 |  | 4 | 125 |
|  | 4 | 250 | 2 | 5 | 100 |
| 1 | 5 | 200 |  | 10 | 50 |
|  | 8 | 125 |  | 20 | 25 |
|  | 10 | 100 |  | 5 | 40 |
|  | 20 | 50 | 5 | 8 | 25 |
|  | 25 | 40 |  | 10 | 20 |
| 4 | 5 | 50 |  |  |  |
|  | 10 | 25 | 10 | 10 | 10 |

Solution 2: We can find all these numbers by listing the powers of 2 and 5, without forgetting that one of the dimensions can be 1 (in the case of power 0). The table shows the 19 possibilities for these blocks.

| power of 2 | power of 5 | $a$ | $l$ | $c$ |
| :---: | :---: | :---: | :---: | :---: |
| 3 | 3 | 1 | 1 | $2^{3} \times 5^{3}$ |
|  |  | 1 | $2^{3}$ | $5^{3}$ |
| 1,2 | 3 | 1 | 2 | $2^{2} \times 5^{3}$ |
|  |  | 1 | $2^{2}$ | $2 \times 5^{3}$ |
|  |  | 2 | $2^{2}$ | $5^{3}$ |
| $1,1,1$ | 3 | 2 | 2 | $2 \times 5^{3}$ |
| 3 | 1,2 | 1 | $2^{3} \times 5$ | $5^{2}$ |
|  |  | 1 | $2^{3} \times 5^{2}$ | 5 |
|  |  | $2^{3}$ | 5 | $5^{2}$ |
| 3 | $1,1,1$ | 5 | 5 | $2^{3} \times 5$ |
| 1,2 | 1,2 | 1 | $2 \times 5$ | $2^{2} \times 5^{2}$ |
|  |  | 1 | $2 \times 5^{2}$ | $2^{2} \times 5$ |
|  |  | 2 | 5 | $2^{2} \times 5^{2}$ |
|  |  | $2^{2}$ | $2 \times 5$ | $5^{2}$ |
|  |  | $2^{2}$ | $2 \times 5^{2}$ | 5 |
| 1,2 | $1,1,1$ | 2 | $2^{2} \times 5$ | $5^{2}$ |
| $1,1,1$ | 1,2 | 5 | $2 \times 5$ | $2^{2} \times 5$ |
| $1,1,1$ | $1,1,1$ | $2 \times 5$ | $2 \times 5$ | $2 \times 5^{2}$ |