3.33 If prime numbers $p, q$ satisfy the equation $3 p+5 q=31$, what is the value of $\log _{2} \frac{p}{3 q+1}$?
(China Junior High School Mathematics League, 1988)
[Solution] From the given equation, we know that one of $3 p$ and $5 q$ must be even, and the other must be odd.
If $3 p$ is even, then $p=2$, and thus $q=5$. In this case,
$$
\log _{2} \frac{p}{3 q+1}=\log _{2} \frac{2}{3 \cdot 5+1}=\log _{2} \frac{1}{8}=-3 ;
$$

If $5 q$ is even, then $q=2$, and thus $p=7$. In this case,
$$
\log _{2} \frac{p}{3 \cdot q+1}=\log _{2} 1=0 \text {, }
$$

Therefore,
$$
\log _{2} \frac{p}{3 q+1}=-3 \text { or } 0 \text {. }
$$