What is the smallest positive integer that has exactly eight distinct positive factors?
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. 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. 