Question

How many pairs of positive integers (a,b), where a≤b satisfy 1/a+1/b=1/50?

Answer 1

well, you know that (100,100) is one pair. So, there are a maximum of 99 other pairs that will fit. That's a lot to check by hand, so there must be some theoretical way to resolve them.

1/4 + 1/4 = 1/2
1/6 + 1/3 = 1/2

1/6 + 1/6 = 1/3
1/12 + 1/4 = 1/3

1/8 + 1/8 = 1/4
1/12 + 1/6 = 1/4
1/20 + 1/5 = 1/4

Still looks tedious to run down the list starting at 1/100

Answer 2

1/b = 1/50 - 1/a = (a-50)/50a
b = 50a/(a-50)

If a = 100, b = 5000/50 = 100
If a = 75, b = 150
There are no other integer a values ¡Üb that yield an integer for b.

50/(a-50) would have to be an integer

Answer 3

well, what the correct answer?