113. For what least natural $n$ is each of the fractions

$$
\frac{7}{n+9}, \frac{8}{n+10}, \ldots, \frac{31}{n+33}
$$

irreducible

146
Solution. All the fractions have the form $\frac{k}{k+(n+2)}$, and such a fraction will be irreducible if the numbers $k$ and $n+2$ have no common divisors. Thus, $n+2$ must be coprime with the numbers $7,8, \ldots, 31$, and the smallest natural number with this property is 37. Therefore, the desired number $n$ is 35.

Answer: 35.