How many ordered pairs of positive integers , satisfy the following equation?
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Rearranging
a b − 1 2 0 ( a + b ) = 0
a b − 1 2 0 ( a + b ) + 1 2 0 2 = 1 2 0 2
( a − 1 2 0 ) ( b − 1 2 0 ) = 1 2 0 2
Set x = ( a − 1 2 0 ) and y = ( b − 1 2 0 )
x y = 1 2 0 2
Solving for possible values of x, we see that over the integers, x must be a factor of 1 2 0 2 . If we factor 120, we get 1 2 0 = 2 3 ⋅ 3 ⋅ 5
∴ 1 2 0 2 = 2 6 ⋅ 3 2 ⋅ 5 2 .
This has 7 ⋅ 3 ⋅ 3 factors ( explained here )
Therefore, our answer is 63.