2. In the notebook, all irreducible fractions with the numerator 15 are written down, but which are greater than $\frac{1}{16}$ and less than $\frac{1}{15}$. How many such fractions are written in the notebook?
Answer: 9 fractions.

## Solution:

We are looking for all suitable irreducible fractions of the form $\frac{15}{n}$. Since $\frac{1}{16}<\frac{15}{n}<\frac{1}{15}$, then $15 \cdot 15<n<15 \cdot 16$ or $225<n<240$ (with the fraction $\frac{15}{n}$ being irreducible, meaning $n$ is not divisible by 3 or 5). It is not difficult to verify that from this range (of 14 numbers), we need to exclude the numbers 228, 230, 231, 234, 235, 237 - 6 numbers. This leaves 9 possible values for $n$.

## Criteria:

Correct answer without a correct justification - 1 point.

Participants might try to list all such fractions and then justify why the denominator cannot be larger or smaller - carefully monitor the accuracy of such justifications.