# Task № 5.1

## Condition:

Polina thought of a natural number. Her friends asked one question each:

Masha: Is it divisible by 11?

Irina: Is it divisible by 13?

Anya: Is it less than 15?

Olya: Is it divisible by 143?

Polina answered affirmatively to only two of the four questions. What numbers could Polina have thought of? List all possible options.

## Answers:

$\circ 11$

$\circ 13$

## Exact match of the answer - 1 point
Solution.

If Olya were right, the number would be divisible by 143, but in that case, it would also be divisible by 11 and 13, which means there would be at least three affirmative answers. Therefore, the number is not divisible by 143. If both Masha and Irina were right, the number would be divisible by both 11 and 13, but in that case, it would also be divisible by 143, which we have already determined is not the case. Therefore, at most one of Masha's and Irina's answers could be affirmative. In order for there to be two affirmative answers, Anya must be right, and one of Masha and Irina must be wrong. This means there are two possibilities: either Masha and Anya are right, or Irina and Anya are right. In the first case, the answer is 11, and in the second case, it is 13.