2021 April MIMC 10 Problems/Problem 11
How many factors of is a perfect cube or a perfect square?
We need to calculate the number of perfect squares and the number of perfect cubes and then subtract the number of th power according to the principle of inclusion and exclusion. First of all, we need to factor . Since we can choose even amount of each factor, there are a total of perfect squares. Using the same logic, any number that is a cube must have multiple of factors for each factor. Therefore, there are cubes. In addition, there are numbers with th power. In total, there are perfect square or perfect cube factors of .