30. It is given that $a$ and $b$ are positive integers such that $a$ has exactly 9 positive divisors and $b$ has exactly 10 positive divisors. If the least common multiple (LCM) of $a$ and $b$ is 4400 , find the value of $a+b$.
30. Answer: 276
Since the number of positive divisors of $a$ is odd, $a$ must be a perfect square. As $a$ is a divisor of $4400=2^{4} \times 5^{2} \times 11$ and $a$ has exactly 9 positive divisors, we see that $a=2^{2} \times 5^{2}$. Now the least common multiple of $a$ and $b$ is 4400 implies that $b$ must have $2^{4} \times 11$ as a divisor. Since $2^{4} \times 11$ has exactly 10 positive divisors, we deduce that $b=2^{4} \times 11=176$. Hence $a+b=276$.