. Find the smallest integer $n>0$ for which the fractions:

$$
\frac{19}{n+21}, \frac{20}{n+22}, \frac{21}{n+23}, \ldots, \frac{91}{n+93}
$$

are all irreducible.
. Note that the difference between the numerator and the denominator of each fraction is $n+2$. Therefore, $n+2$ must be coprime with all integers from 19 to 91. Since this list includes all prime numbers less than or equal to 91, $n+2$ must have only prime factors strictly greater than 91. The smallest integer satisfying this condition is 97, so $n=$ 95.