2. Find the fraction of the form $\frac{n}{23}$ that is closest to the fraction $\frac{37}{57}$ ( $n$ - integer).
Solution. After bringing to a common denominator, we need to find the smallest deviation of the fraction $\frac{57 n}{1311}$ from $\frac{851}{1311}$. Dividing 851 by 57 with a remainder gives $851=14 \cdot 57+53$. If $n=14$, then the deviation of the number $57 n$ from 851 is 53, and for $n=15$ this deviation is 4. For all other $n$, its value is greater than 53. Thus, the required $n=15$, and the fraction is $\frac{15}{23}$.

Answer: $\frac{15}{23}$.