Here's the exact wording of the problem:
"The squares, of course, are the numbers 1,4,9,... The square-free numbers are the integers 1,2,3,5,6,... which are not divisible by the square of any prime (so that 1 is both square and square-free). Show that every positive integer is uniquely representable as the product of a square and a square-free number. Show that there are infinitely many square-free numbers."
I was able to prove the first statement with a proof by contradiction, but can't figure out the second part. Does it rely on the first part?