parmenides51 wrote: Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible. We can express all these fractions as $\frac{a}{a+n+2}$ and note that this is irreducible if: $$a \not{\mid} \; a+n+2 \implies a \not{\mid} \; n+2 \implies \gcd(a,n+2)=1$$Since $a \in (19, 91)$ the the least number $n+2$ such is $97$ thus $n=95$. And we are done