For the equation: 1/x + 1/y = 1/7,
I believe that there are 3 positive integer solutions: (8, 56) (14, 14) and (56, 8).
Adding 8 + 14 + 56 = 78.
My analysis: 1/x + 1/y = 1/7 ---> multiplying by 7xy ---> 7x + 7y = xy
Solving for y: 7y - xy = -7x ---> y(7 - x) = -7x ---> y = (-7x) / (7 - x) ---> y = (7x) / (x - 7)
Using this equation, as x gets larger and larger, y approaches 7, but is always greater than 7.
Similarly, solving for x: x = (7y) / (y - 7)
So, as y gets largr and larger, x approaches 7, but is always greater than 7.
So, 8 is the smallest possible value for either x or y.
When x = 8, y = 56.
This makes 56 the largest possible value.
Trying values between 8 and 56 gives the other solution, 14.
(This solution can be determined by a different analysis.)