## Task B-2.4.

Matko wrote down five different numbers on a piece of paper, and when Ana asked what the numbers were, he said: "If you choose any four of the five written numbers, their product will be one of the numbers 9, 12, 18, 24, 36." What numbers did Matko write down?
## Solution.

Let the recorded numbers be $a, b, c, d, e$. Then we have

$$
\begin{aligned}
a b c d & =9 \\
a b c e & =12 \\
a b d e & =18 \\
a c d e & =24 \\
b c d e & =36
\end{aligned}
$$

If we multiply these four equations, we get

$$
\begin{aligned}
& a^{4} \cdot b^{4} \cdot c^{4} \cdot d^{4} \cdot e^{4}=9 \cdot 12 \cdot 18 \cdot 24 \cdot 36 \\
& 1 \text { point } \\
& (a b c d e)^{4}=3^{2} \cdot 3 \cdot 2^{2} \cdot 2 \cdot 3^{2} \cdot 3 \cdot 2^{3} \cdot 2^{2} \cdot 3^{2}=2^{8} \cdot 3^{8}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_30_bcbf631b6726be7fa637g-12.jpg?height=43&width=108&top_left_y=344&top_left_x=1759)

$$
\begin{aligned}
& a b c d e=4 \cdot 9=36 \\
& (* *) \quad 1 \text { point }
\end{aligned}
$$

By comparing $(*)$ and ( $* *)$, we get

$$
a b c d e=9 e=36 \quad \Longrightarrow \quad e=4
$$

and similarly, we obtain

$$
d=\frac{36}{12}=3, c=\frac{36}{18}=2, b=\frac{36}{24}=\frac{3}{2}, a=\frac{36}{36}=1
$$

The sought numbers are $1, \frac{3}{2}, 2, 3$ and 4.