Suppose that $b$ is a positive integer greater than or equal to $2.$ When $197$ is converted to base $b$, the resulting representation has $4$ digits. What is the number of possible values for $b$?
So when we convert $197$ to base $b$, we get $4$ digits. That means $b^3 \leq 197 < b^4$. Right. Now we just need to find the number of possible values for $b$. Well we know $2 \leq b \leq \sqrt[3]{197}$. And by doing some calculations we can find that $\sqrt[3]{197} \approx 6.3$, and so $2 \leq b \leq 6$. Now we can see that $b$ can only be $4, 5$. Right. So there are only $2$ possible values for $b$.

