Question 24
For all quadruples $(x, y, z, w)$ consisting of integers $1 \le x, y, z, w \le 1000$, we consider adding up the maximum value of $xy + zw$, $xz + yw$, $xw + yz$ and denote the sum by $M$. Similarly, for all quadruples $(x, y, z, w)$ consisting of integers $1 \le x, y, z, w \le 1000$, we consider adding up the minimum value of $xy + zw$, $xz + yw$, $xw + yz$ and denote the sum by $m$. Determine the number of positive divisors of $M - m$.