Example 1. Find all integer values of \( l \) that make the expression \(\frac{5l+6}{8l+7}\) reducible. (19th Moscow Mathematical Olympiad)
To find all integer values of $l$ that satisfy $(5 l+6,8 l+7)>1$. Because
$$
\begin{array}{l}
(5 l+6,8 l+7)=(5 l+6,3 l+1) \\
=(3 l+1,(-2)(3 l+1)+5 l+6) \\
=(3 l+1, l-4) \\
=(3 l+1+(-3)(l-4), l-4) \\
=(13, l-4)>1
\end{array}
$$

From $13 \mid l-4$, we get $l=13 m+4(m \in \mathbb{Z})$.