Question

https://youtube.com/shorts/RWWLp6zpmHI?si=hru5AImqfciy_8A0 use this trick i got 15 and taking as pair will be 15 plus 15 as 30 subtract 1 get 29..

You can refer to the playlist below to get a better understanding of such qns.

https://www.youtube.com/watch?v=oFDNfUNc8l0&list=PLVVVfitxrCHxItC3h6N5wF9R6XVJT-21b

the equation is

1/a + 1/b = 1/99

first combine the fractions

(a + b) / ab = 1 / 99

now cross multiply

99(a + b) = ab

rearrange

ab - 99a - 99b = 0

now use a small algebra trick - add 99^2 to both sides so it factors nicely

ab - 99a - 99b + 99^2 = 99^2

this becomes

(a - 99)(b - 99) = 99^2

and

99^2 = 9801

so now the question becomes - how many integer pairs multiply to 9801?

prime factorize 99

99 = 3^2 x 11

so

99^2 = 3^4 x 11^2

number of positive factors = (4 + 1)(2 + 1) = 5 x 3 = 15

each factor pairs with another factor - so 15 positive pairs

since negative x negative = positive, we also get 15 negative pairs

total = 30

but one case gives a = 99 and b = 99, which does not work in the original setup

so final answer = 29