solve for all integer solutions to the diophantine equation

Question

$$1/x+1/y=1/14$$

Find all integer solutions for x and y.

I can solve linear diophantine equations without a problem normally but this has me stumped.

Answer 1

The equation gives: $14y = xy - 14x$. So solving for $x$ and get: $x = 14 + \frac{196}{y-4}$.

So $x$ is an integer if $y - 4$ divides $196$. We can take it from here....

Answer 2

Rewrite as $xy-14x-14y=0$, and then, in an analogue of completing the square, as $(x-14)(y-14)=196$.

So $x-14$ ranges over the divisors of $196$. Since $196=2^2\cdot 7^2$, $196$ has $(2)(3)(3)$ integer divisors, including the negative divisors. That gives $18$ possible values of $x$.