4. How many three-digit numbers less than 200 can be expressed in the form $a^{2} \cdot b$, where $a$ and $b$ are prime numbers?
4. If $a=2$, then $a^{2}=4$. For the product to be a three-digit number, $b$ must be a prime number greater than or equal to 25. For the product to be less than 200, $b$ must be a prime number less than 50.

Then $b \in\{29,31,37,41,43,47\}$.

1 POINT

If $a=3$, then $a^{2}=9$, so $b$ needs to be a prime number greater than 11 and less than or equal to 22.

Then $b \in\{13,17,19\}$.

1 POINT

For $a=5$, $a^{2}=25$, so $b \in\{5,7\}$.

1 POINT

For $a=7$, $a^{2}=49$, so $b \in\{3\}$.

1 POINT

For $a \geq 11$, $a^{2} \cdot b>200$ for any prime number $b$, so there are no more solutions.

Therefore, the total number of solutions is $6+3+2+1=12$.