Number Theory Problem by Ashu Dablo

Number Theory Level 5

$\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{580909190400000}$ has $n$ ordered pairs of positive integral solutions .

Find $n$ $\mod (2\times3\times5\times7)$

Note: This problem can be solved with a little patience and pencil and paper and so, please refrain from using computational aids.