On the paper, there were several consecutive positive multiples of a certain natural number greater than one. Radek pointed to one of the written numbers: when he multiplied it by the number that was adjacent to it on the left, he got a product that was 216 less than when he multiplied it by the number that was adjacent to it on the right.

Which number could Radek have pointed to? Find all possible options.

(L. Šimünek)
Let the natural number whose multiples were written on the paper be denoted by $n$; according to the problem, $n>1$. The numbers on the paper can thus be denoted as

$$
(k-1) n, \quad k n, \quad(k+1) n,
$$

where the unknown $k$ is a natural number; for all three expressions to be positive, $k>1$. Radek thus received the products $(k-1) k n^{2}$ and $(k+1) k n^{2}$, whose difference is $2 k n^{2}$. It is given that $2 k n^{2}=216$, which simplifies to

$$
k n^{2}=108=2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \text {. }
$$

Respecting the conditions $n>1$ and $k>1$, we get three different solutions:

| $n$ | 2 | 3 | 6 |
| :---: | :---: | :---: | :---: |
| $k$ | 27 | 12 | 3 |
| $k n$ | 54 | 36 | 18 |

There are three possible numbers that Radek could have pointed to: 54, 36, or 18.

Scoring suggestion. 2 points for the equation $k n^{2}=108=2^{2} \cdot 3^{3}$ or its equivalent; 1 point for each solution that meets the problem's conditions; 1 point for a correctly formulated conclusion.

Listing the three terms of the sequence is not a necessary part of the solution. For clarity, we list them: a) $52,54,56$; b) $33,36,39$; c) $12,18,24$.