Japan Mathematical Olympiad

$$\boxed{20412}$$
For 3 real numbers $a, b, c$, the difference of the maximum value and the minimum value of them is $\frac{|a-b|+|b-c|+|c-a|}{2}$. Using $|(xy + zw) - (xz + yw)| = |x - w||y - z|$, we have
$$M - m = \frac{1}{2} \sum_{x,y,z,w=1}^{1000} \left( |x - w||y - z| + |x - y||z - w| + |x - z||y - w| \right).$$
Since
$$\sum_{x,y,z,w=1}^{1000} |x - w||y - z| = \left( \sum_{x,y=1}^{1000} |x - y| \right)^2 = \left( 2 \sum_{d=1}^{999} d(1000 - d) \right)^2 = \frac{(999 \cdot 1000 \cdot 1001)^2}{9},$$
we have
$$M - m = \frac{3}{2} \cdot \frac{(999 \cdot 1000 \cdot 1001)^2}{9} = 2^5 \cdot 3^5 \cdot 5^6 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 37^2.$$
Therefore, the number of positive divisors of $M - m$ is $6 \cdot 6 \cdot 7 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 20412$.