8.1. The hikers had several identical packs of cookies. At the midday break, they opened two packs and divided the cookies equally among all the participants of the hike. One cookie was left over, and the hikers fed it to a squirrel. At the evening break, they opened another three packs and also divided the cookies equally. This time, 13 cookies were left over. How many hikers were on the hike? Justify your answer.
Solution: Let there be $n$ tourists in the group, and $m$ cookies in each pack (where $m$ and $n$ are natural numbers, and $n > 13$). According to the problem, the numbers $x = 2m - 1$ and $y = 3m - 13$ are divisible by $n$. Therefore, the number $3x - 2y = 23$ is also divisible by $n$. Since 23 is a prime number and has only two natural divisors, the divisor 1 is not suitable because it is less than 13. Thus, $n = 23$.

We also need to verify that this situation is possible. So, let's assume there are 23 tourists in the group. Then the described situation will occur, for example, if there are 12 cookies in each pack.

Answer: There are 23 tourists in the group.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| It is proven that there could only be 23 tourists, and an example of the number of cookies in a pack that satisfies the condition of the problem is provided | 7 points |
| It is proven that there could only be 23 tourists, but no example of the number of cookies in a pack that satisfies the condition of the problem is provided | 5 points |
| A correct example (with 23 tourists) is provided, but it is not proven that there cannot be a different number of tourists | 2 points |
| The correct answer without justification and an example | 0 points |