### Instruction: What is the smallest positive integer that has exactly eight distinct positive factors?### Response: To find the smallest positive integer with exactly eight distinct positive factors, I need to think about how the number of factors of a number is related to its prime factorization.
If a number n can be written as the product of prime powers, n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}, where p_1, p_2, \cdots, p_k are distinct primes and a_1, a_2, \cdots, a_k are positive integers, then the number of factors of n is given by the formula (a_1 + 1)(a_2 + 1) \cdots (a_k + 1).
This is because for each prime power p_i^{a_i}, I can choose any exponent from 0 to a_i to form a factor of n, and there are a_i + 1 choices for each exponent.
So, to find the smallest positive integer with exactly eight factors, I need to find the smallest way to write 8 as the product of factors of the form a_i + 1.
One way to do this is to write 8 as 2 \times 2 \times 2, which means I can choose three distinct primes and raise them each to the first power.
The smallest three distinct primes are 2, 3, and 5, so one possible answer is n = 2^1 3^1 5^1 = 30.
But is this the smallest possible answer?
Another way to write 8 as a product of factors of the form a_i + 1 is to write 8 as 4 \times 2, which means I can choose one prime and raise it to the third power, and another prime and raise it to the first power.
The smallest way to do this is to choose 2 and 3, so another possible answer is n = 2^3 3^1 = 24.
This is smaller than 30, so I have found a better answer.
But is this the smallest possible answer?
The only other way to write 8 as a product of factors of the form a_i + 1 is to write 8 as 8 \times 1, which means I can choose one prime and raise it to the seventh power.
The smallest way to do this is to choose 2, so another possible answer is n = 2^7 = 128.
This is larger than 24, so I can ignore this answer.
Therefore, the smallest positive integer with exactly eight distinct positive factors is 24.