How can I solve equations of the form $xy-k(x+y)=0$ where $k\geq 0$ where $x,y\in \mathbb{Z}$

Question

Given $xy-k(x+y)=0$ where $k\geq 0\space where\space x,y\in \mathbb{Z} $

I know this is a diophantine equation which I have read about earlier.

My attempt :

$(x-k)\cdot(y-k)=k^2$

$\implies \space (x,y) \in {all \space factors \space of \space k^2}$

I am having some doubt about this approach but this proved to be right in $xy-6(x+y)=0$

I know this is a very silly and easy question but please do care to throw some light.

Answer 1

Your method is good.

$xy-6(x+y)=0$ $\implies$ $(x-6)(y-6)=36$

$(x-6,y-6)=(1,36),(2,18),(3,12),(4,9),(6,6),(9,4),(12,3),(18,2),(36,1)(-1,-36),(-2,-18),(-3,-12),(-4,-9),(-6,-6),(-9,-4),(-12,-3),(-18,-2),(-36,-1)$

$(x,y)=(7,42),(8,24),\dots$