Anetka's uncle has his birthday on the same day of the year as Anetka's aunt. The uncle is older than the aunt, but not by more than ten years, and both are adults. At the last celebration of their birthdays, Anetka realized that if she multiplied their celebrated ages and then multiplied the resulting product by the number of dogs that gathered at the celebration, she would get the number 2024.

How many dogs could have been at this celebration? Determine all possibilities.

(M. Petrová)
The number 2024 needs to be expressed as the product of three natural numbers, two of which are greater than or equal to 18 and differ by no more than 10. We will consider the possible options based on the prime factorization of the number 2024, which is as follows:

$$
2024=2 \cdot 2 \cdot 2 \cdot 11 \cdot 23 \text {. }
$$

The prime numbers 11 and 23 belong to the ages of the aunt and uncle, each to a different person; if this were not the case, it would not be possible to form two factors greater than 18 from the given prime numbers. Moreover, none of the ages can be greater than $2 \cdot 23=46$ years; the next higher possible age is $2 \cdot 2 \cdot 23=92$ years, and in such a case, it would not be possible to form a factor from the given prime numbers that differs from 92 by no more than 10. Therefore, it is sufficient to consider the following options:

- If the aunt or uncle were 23 years old, the other would have to be between 18 and 33 years old. Of the possible factors, only $2 \cdot 11=22$ meets this requirement. Thus, the aunt could be 22 and the uncle 23 years old. In this case, there would be $2 \cdot 2=4$ dogs at the party.
- If the aunt or uncle were $2 \cdot 23=46$ years old, the other would have to be between 36 and 56 years old. Of the possible factors, only $2 \cdot 2 \cdot 11=44$ meets this requirement. Thus, the aunt could be 44 and the uncle 46 years old. In this case, there would be 1 dog at the party.

At the party, there was either one dog or four dogs.

Note. The product of the ages of the aunt and uncle was greater than $18 \cdot 18=324$. Therefore, the number of dogs at the party was at most the integer part of the quotient 2024 : 324, which is 6. The given prime numbers can only form the numbers 1, 2, and 4. The analysis of the possibilities can be based on these three cases:

- Cases with 1 or 4 dogs are possible, see the previous solution.
- The case with 2 dogs is not possible. The product of the ages of the aunt and uncle would be $2 \cdot 2 \cdot 11 \cdot 23$, which can be expressed as the product of two factors greater than 18 either as $22 \cdot 46$ or $44 \cdot 23$. However, in both decompositions, the difference between the factors is greater than 10.