If $a^{2}+3 b=33$, where $a$ and $b$ are positive integers, what is the value of $a b$ ?
(A) 11
(B) 24
(C) 16
(D) 32
(E) 27
We try positive integer values for $a$ to see the resulting value of $b$ is a positive integer.

If $a=1$, then $a^{2}=1$, so $3 b=32$, so $b$ is not a positive integer.

If $a=2$, then $a^{2}=4$, so $3 b=29$, so $b$ is not a positive integer.

If $a=3$, then $a^{2}=9$, so $3 b=24$, so $b=8$.

If $a=4$, then $a^{2}=16$, so $3 b=17$, so $b$ is not a positive integer.

If $a=5$, then $a^{2}=25$, so $3 b=8$, so $b$ is not a positive integer.

If $a$ is at least 6 , then $a^{2}$ is at least 36 , so $3 b$ is negative, and $b$ cannot be a positive integer. Therefore, there is only one possible pair of values for $a$ and $b$, so $a b=3(8)=24$.

Answer: (B)