Problem 5.  Let N denote the set of positive integers. Find all functions f : N → N such that the following conditions are true for every pair of positive integers ( x, y ):
(i) x and f ( x ) have the same number of positive divisors.
(ii) If x does not divide y and y does not divide x , then

gcd( f ( x ) , f ( y )) > f (gcd( x, y )) .

Here gcd( m, n ) is the largest positive integer that divides both m and n .