Find all integers x for which there exists an integer y such that 1/(x)+1/(y)=1/7 (in other words, find all ordered pairs of integers (x, y) that

1/x +  1/y  = 1/7   simplify

[x + y ] / xy  =  1/7

7[x + y]   = xy

7x + 7y - xy   = 0          rewrite as

xy - 7x - 7y   = 0        add 49 to both sides

xy - 7x - 7y +  49  =  49      factor

(x - 7) ( y - 7)   = 49       (1)

Note that the following  ordered pairs of integers would work :

(14, 14)

(8, 56)

(56, 8)

(-42, 6)

(6, -42)

Note that (0,0)   would  also satisfy (1), but  ......  x and y cannot be 0 in the original problem  [division by 0 ]

An oldie but a goodie.

Thanks Chris,

And Thanks Heureka for bringing me back here.