[Example 4.5.6] Given that $p, q$ are both integers greater than 1, and $2 p-1$ is a multiple of $q$, $2 q-1$ is a multiple of $p$, find the value of $p+q$.
The problem does not provide specific numbers, yet it requires finding the value of $p+q$. This presents a certain difficulty. If we analyze the given conditions one by one, we almost get no useful information. However, if we look at the conditions as a whole, from the fact that $2 p-1$ is a multiple of $q$ and $2 q-1$ is a multiple of $p$, we can observe the integers $\frac{2 p-1}{q}$ and $\frac{2 q-1}{p}$ uniformly. We can then discover that at least one of $\frac{2 p-1}{q}$ and $\frac{2 q-1}{p}$ must be less than 2. Otherwise, if
$$
\frac{2 p-1}{q} \geqslant 2, \frac{2 q-1}{p} \geqslant 2
$$

were both true, then we would have
$$
\left\{\begin{array}{l}
2 p-1 \geqslant 2 q, \\
2 q-1 \geqslant 2 p
\end{array}\right.
$$

Adding these inequalities gives
$$
-2 \geqslant 0 \text {. }
$$

This is impossible.
By observing the given conditions as a whole, we have found a way to solve the problem.
Since at least one of $\frac{2 q-1}{p}$ and $\frac{2 p-1}{q}$ is less than 2, let's assume $\frac{2 p-1}{q} < 2$. Then, we have
$$
2 p - 1 < 2 q \implies 2 p - 1 = q.
$$

Since $q$ is a positive integer, $2 p - 1$ must be a positive integer, and $p$ must be an integer. The only solution for $p$ is $p=3$.
At this point,
$$
q=2 p-1=5.
$$

Thus,
$$
p+q=8.
$$