Problem 11.1. The teacher wrote a two-digit number on the board. Each of the three boys made two statements.

- Andrey: “the number ends with the digit 6” and “the number is divisible by 7”.
- Borya: “the number is greater than 26” and “the number ends with the digit 8”.
- Sasha: “the number is divisible by 13” and “the number is less than 27”.

It is known that each boy was right once and wrong once. What number could have been written on the board? List all possible options.
Answer: 91.

Solution. Let's consider Sasha's statements. If his second statement that the number is less than 27 is true, then Borya's first statement is definitely false, so the number must end in 8. The only two-digit number that meets these conditions is 18, but then none of Andrei's statements are true. Contradiction.

Therefore, the first statement of Sasha that the number is divisible by 13 must be true; then his second statement is false, and the number is not less than 27. But in this case, Borya's first statement that the number is greater than 26 is true, and his second statement that the number ends in 8 is false. Among the two-digit numbers divisible by 13, greater than 26, and not ending in 8, only the number 91 fits (the numbers $39,52,65$ do not fit because both of Andrei's statements would be false).