Answer: To solve the equation 1/x + 1/y = 1/19 for integer solutions, we can use a common method called "diophantine equations" which involves finding solutions to equations with integer variables.
Multiplying both sides by the least common multiple of x, y, and 19, we get:
19y + 19x = xy
Bringing all the terms to one side, we have:
xy - 19x - 19y = 0
Using Simon's Favorite Factoring Trick, we can add 361 to both sides of the equation, which gives:
xy - 19x - 19y + 361 = 361
(x - 19)(y - 19) = 361
Now, we need to find all the pairs of integers (x, y) such that (x - 19)(y - 19) = 361.
The factors of 361 are 1, 19, and 361 itself. So, we can solve the equation by setting x - 19 equal to each factor and finding the corresponding value of y:
x - 19 = 1, y - 19 = 361, then x = 20, y = 380
x - 19 = 19, y - 19 = 19, then x = 38, y = 38
x - 19 = 361, y - 19 = 1, then x = 380, y = 20
Therefore, the integer solutions to the equation 1/x + 1/y = 1/19 are (x, y) = (20, 380), (38, 38), and (380, 20).
Explanation: