Example 1. Find the smallest positive integer $n$ that makes $\frac{n-13}{5 n+6}$ a non-zero reducible fraction.
Solution: Since $(1,5)=1,|1 \times 6-(-13) \times 5|$ $=71 \neq(1,5),(1,71)=1$, the fraction can be simplified by 71. From $\left.n-13=71 m_{1}, n=71 m_{1}+13\left(m_{1} \in Z\right)\right)$, and $n-13>0$, so we take $m_{1}=1$, obtaining the smallest $n=84$.