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.

1 solution

Ashu Dablo
Oct 6, 2014

This equation can be rearranged and written as

(580909190400000-x)(58090919040000-y)= $58090919040000^{2}$

58090919040000= $2^{11}*3^{7}*5^{5}*7^{3}*11^{2}$

So no. of factors of $58090919040000^{2}$ are $23*15*11*7*5=132825$ . Number of ordered pairs are also equal to 132825.

132825= 1 mod 2

132825= 0 mod 3

132825= 0 mod 5

132825= 0 mod 7

So it is equal to 105 mod 210.

Answer is $\boxed{105}$