What is the smallest positive integer with exactly 10 positive integer divisors?
To have exactly 10 positive integer divisors, the number must be of the form p^4 q or p^9, where p and q are distinct prime numbers. This is because the number of divisors of a product of prime powers is the product of one more than the exponents. For example, p^4 q has (4 + 1)(1 + 1) = 10 divisors, and p^9 has (9 + 1) = 10 divisors. The smallest possible value of p is 2, and the smallest possible value of q is 3. So, I need to compare 2^4 3 and 2^9 and see which one is smaller. 2^4 3 = 16 3 = 48, and 2^9 = 512. Clearly, 48 is smaller than 512, so the smallest positive integer with exactly 10 positive integer divisors is 48. 