3. Determine the smallest natural number $n$ for which none of the fractions

$$
\frac{7}{n+9}, \frac{8}{n+10}, \frac{9}{n+11}, \ldots, \frac{2015}{n+2017}
$$

can be simplified.
3. All the given fractions are of the form $\frac{k}{k+n+2}$. They cannot be simplified if and only if $\gcd(k, k+n+2)=1$, which is true if and only if $\gcd(k, n+2)=1$. Therefore, we are looking for the smallest natural number $n$ such that $n+2$ is coprime with $7, 8, 9, \ldots, 2015$. For $1 \leqslant n \leqslant 4$, it is easy to find a number in the observed sequence with which $n+2$ is not coprime. For $5 \leqslant n \leqslant 2013$, we have that $n+2$ is exactly equal to some member of the observed sequence (and thus is not coprime with it). For $n=2014$, we have $n+2=2016$, which is an even number, and obviously not coprime with, for example, 8. However, for $n=2015$, we have $n+2=2017$, and since 2017 is a prime number, it is coprime with all members of the observed sequence, so the solution to the problem is $n=2015$.