8.1. In the class, there are more than 20 but fewer than 30 students. In the class, those who attend the chess club are half the number of those who do not attend. And those who attend the checkers club are one-third the number of those who do not attend. How many students are in the class? Provide all possible answers and prove that there are no others.
Solution: Let $n$ be the number of students in the class who attend the chess club, then $2n$ students do not attend, and the total number of students in the class is $3n$, meaning the total number of students in the class is divisible by 3. Similarly, from the fact that the number of people attending the checkers club is three times less than those who do not attend, we get that the total number of students in the class is divisible by 4. The numbers 3 and 4 are coprime, so the total number of students in the class must also be divisible by 12. In the range from 20 to 30, there is only one such number: 24. It is easy to construct an example showing that a class can contain 24 students: in this case, 8 people attend the chess club, and 6 attend the checkers club.

Answer: 24 students.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer (an example that 24 students can be in the class is NOT REQUIRED) | 7 points |
| Proven that the total number of students in the class is divisible by both the number 3 and the number 4 | 5 points |
| Proven that the total number of students in the class is divisible by one of the numbers 3 or 4 with no further progress | 3 points |
| The correct answer is given, supported by an example, and the uniqueness is not proven | 1 point |
| Answer without justification and/or incorrect answer | 0 points |