Each brick in the following pyramid contains one number. Whenever possible, the number in each brick is the least common multiple of the numbers in the two bricks directly below it.

What number can be in the bottom brick? Determine all possibilities.

(A. Bohiniková)

![](https://cdn.mathpix.com/cropped/2024_04_17_b1dd73eb0c32fed574d9g-5.jpg?height=354&width=554&top_left_y=1596&top_left_x=751)

Hint. What is the least common multiple of three numbers, one of which is a divisor of another?
The number 30 has a total of 8 divisors, which can be substituted for $A$ and $B$.

![](https://cdn.mathpix.com/cropped/2024_04_17_b1dd73eb0c32fed574d9g-5.jpg?height=340&width=548&top_left_y=2234&top_left_x=754)

The number $C$ is the least common multiple of 13 and $A$, the number $E$ is the least common multiple of $C$ and 30, i.e., $E$ is the least common multiple of the numbers $13, A$, and 30. Since $A$ is a divisor of the number 30, $E$ is the least common multiple of the numbers 13 and 30, i.e., $390=13 \cdot 30$.

Similarly, it can be reasoned that regardless of the value of $B$, $F$ is the least common multiple of the numbers $14=2 \cdot 7$ and $30=2 \cdot 3 \cdot 5$, i.e., $210=7 \cdot 30$.

The number $G$ in the bottom brick can therefore only be the least common multiple of the numbers 390 and 210, i.e., $2730=7 \cdot 13 \cdot 30$.

| 13 | $*$ | $*$ | 14 |
| :--- | :--- | :--- | :--- |
| $*$ | 30 | $*$ |  |

Note. If we consider all possible pairs of numbers whose least common multiple is 30, we get a total of 27 possibilities. By filling in the individual cases for $A$ and $B$, everyone will eventually notice that the numbers $E, F$, and thus $G$ are always the same.