we will say that a positive integer is cube-free if there is no integer such that . let be the set of all cube-free integers. we will say that a positive integer is modest if every prime factor of is less than 10 (i.e., 2, 3, 5 or 7). let be the set of all modest integers. what is (i.e., how many cube-free modest integers are there)? (hint: remember that every positive integer greater than 1 has a unique representation as a product of primes.)
