G3.2 Given that $p$ and $q$ are prime numbers satisfying the equation $18 p+30 q=186$. If $\log _{8} \frac{p}{3 q+1}=b \geq 0$, find the value of $b$.
$$
18 p+30 q=186 \Rightarrow 3 p+5 q=31
$$

Note that the number " 2 " is the only prime number which is even.
$3 p+5 q=31=$ odd number $\Rightarrow$ either $p=2$ or $q=2$
If $p=2$, then $q=5 ; b=\log _{8} \frac{p}{3 q+1}=\log _{8} \frac{2}{3 \times 5+1}=\log _{8} \frac{1}{16}<0$ (rejected)
If $q=2$, then $p=7 ; b=\log _{8} \frac{p}{3 q+1}=\log _{8} \frac{7}{3 \times 2+1}=0$