What is the least positive multiple of 72 that has exactly 16 positive factors?
To find the least positive multiple of 72, I need to factorize 72 and see how I can increase the number of factors by adding more prime factors. 72 can be written as 2^3 * 3^2, which means it has (3 + 1) * (2 + 1) = 12 factors. To increase the number of factors by 4, I need to add one more prime factor to the exponent of either 2 or 3, or add a new prime factor. Adding one more prime factor to the exponent of 2 would give me 2^4 * 3^2, which has (4 + 1) * (2 + 1) = 15 factors, which is not enough. Adding one more prime factor to the exponent of 3 would give me 2^3 * 3^3, which has (3 + 1) * (3 + 1) = 16 factors, which is exactly what I want. Therefore, the least positive multiple of 72 that has exactly 16 positive factors is 2^3 * 3^3 = 216. 