8.2. In the notebook, all irreducible fractions with the numerator 15 are written, which are greater than $\frac{1}{16}$ and less than $\frac{1}{15} \cdot$ How many such fractions are written in the notebook?
Answer: 8 fractions.

Solution. We look for all suitable irreducible fractions of the form $\frac{n}{15}$. Since $\frac{1}{16} < \frac{15}{n}$, then $\frac{15}{225} > \frac{15}{n}$, and $n > 225$. Therefore, $225 < n < 240$. The fraction $\frac{n}{15}$ is irreducible, meaning $n$ is not divisible by 3 or 5. It is not difficult to verify that out of the range from 226 to 239 (14 numbers), we need to exclude 6 numbers (228, 230, 231, 234, 235, 237). This leaves 8 possible values for $n$.

## Criteria.

1 point. Correct answer without correct justification.

2 points. Correctly obtained range for $n$.

0 points. Not all cases considered when listing fractions.

Comment. It is quite possible that participants will try to list all such fractions and then justify why the denominator cannot be greater or less, carefully monitoring the accuracy of such justifications.