Example 3 Find the smallest natural number $n$ such that $\frac{n-13}{5 n+6}$ is a non-zero reducible fraction.
(6th IMO)
Solution: Let the common divisor of the numerator and denominator be $d(d>1)$, and let
$$
\begin{array}{l}
n-13=k d, \\
5 n+6=l d .
\end{array}
$$
(2) $-5 \times$ (1) gives
$$
(l-5 k) d=71 \text{. }
$$

Since 71 is a prime number and $d>1$, we have
$$
d=71 \text{ and } l-5 k=1 \text{. }
$$

Thus, $n=13+k \times 71$.
When $k=0$, $n=13$, the numerator is 0, which does not meet the requirement.
Taking $k=1$, $n=84$ is the smallest natural number sought.