2. Two-digit numbers are written on the board. Each number is composite, but any two numbers are coprime. What is the maximum number of numbers that can be written?
Answer: four numbers.

Solution. Evaluation. Since any two of the written numbers are coprime, each of the prime numbers 2, 3, 5, and 7 can appear in the factorization of no more than one of them. If there are five or more numbers on the board, then all prime factors in the factorization of some of them must be at least 11. But this composite number is at least 121. This contradicts the condition that all written numbers are two-digit. Therefore, there are no more than four numbers on the board.

Example. 25, 26, $33, 49$.

Other examples exist.

Grading criteria.

“+” - a complete and justified solution is provided

“ $\pm$ " - a correct estimate is provided, but the example is not given or is incorrect

"耳” - only the correct answer and correct example are provided

“-" - only the answer is provided

“-" - the problem is not solved or is solved incorrectly