An integer is said to be square-free if it is not divisible by the square of any integer g...

An integer is said to be square-free if it is not divisible by the square of any integer greater than 1. Prove the following:

(a) An integer n > 1 is square-free if and only if n can be factored into a product of distinct primes.

(b) Every integer n > 1 is the product of a square-free integer and a perfect square. [Hint: If is the canonical factorization of n, then write ki=2qi + ri where ri, = 0 or 1 according as ki, is even or odd.]