Example 5 Given that $p$ and $q$ are both prime numbers, and that the quadratic equation in $x$
$$
x^{2}-(8 p-10 q) x+5 p q=0
$$

has at least one integer root. Find all pairs of prime numbers $(p, q)$.
(2005, National Junior High School Mathematics Competition)
Explanation: According to the sum of the roots of the equation being \(8 p-10 q\) and the product of the roots being \(5 p q\), if the equation has one positive integer root, then the other root must also be a positive integer. Let's assume the two positive integer roots of the equation are \(x_{1}\) and \(x_{2}\left(x_{1} \leqslant x_{2}\right)\). By the relationship between the roots and coefficients, we have
\[
x_{1}+x_{2}=8 p-10 q, x_{1} x_{2}=5 p q \text{. }
\]

Thus, from \(x_{1} x_{2}=5 p q\), we know that \(x_{1}\) and \(x_{2}\) have the following possible cases:
\[
\left\{\begin{array}{l}
x_{1}=1,5, p, q, 5 p, 5 q, \\
x_{2}=5 p q, p q, 5 q, 5 p, q, p .
\end{array}\right.
\]

Combining \(x_{1}+x_{2}=8 p-10 q\), we can find the pairs of prime numbers \((p, q)=(7,3),(11,3)\).