JEE maths#18

Algebra Level 4

$\large S = \frac{1^{2}}{1\times3} + \frac{2^{2}}{3\times5} + \frac{3^{2}}{5\times7} + \cdots + \frac{500^{2}}{999\times1001}$

For $S$ as defined above, find the number of divisors of $\left \lfloor S \right \rfloor$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.