Given two positive integers a \neq b, let f(a, b) be the smallest integer that divides exactly one of a, b, but not both. Determine the number of pairs of positive integers (x, y), where x \neq y, 1 \leq x, y \leq 100 and \text{gcd}(f(x, y), \text{gcd}(x, y)) = 2.