Find integer solutions of $(1) xy=2x+2y$ and $ (2)xy=2x+y.$

Question

I've tried this for the first one:$xy=2(x+y).$
Therefore either x or y is divisible by 2.
And I'm totally stuck on the second. How to solve these?

Answer 1

Note that $xy=2x+y$ if and only if $xy-2x-y=0$. But $$xy-2x-y=(x-1)(y-2)-2.$$ So we want to solve the equation $$(x-1)(y-2)=2.$$ But it is not easy for the product of two integers to be $2$. The possibilities are:

$x-1=1$, $\quad y-2=2$;

$x-1=2$, $\quad y-2=1$;

$x-1=-1$, $\quad y-2=-2$;

$x-1=-2$, $\quad y-1=-1$.

The other problem can be solved by a very similar method.

Answer 2

Consider the general equation $$axy+bx+cy=d.$$ $$a^2xy+abx+acy+bc=ad+bc$$ $$(ax+c)(ay+b)=ad+bc.$$

For your first example $a=1, b=-2, c=-2, d=0.$
Now $(x-2)(y-2)=4.$
All the solutions of (1) are given by $$x-2=±1,y-2,±4$$ $$x-2=±4,y-2,±1$$ $$x-2=±2,y-2,±2$$

For your first example $a=1, b=-2, c=-1, d=0.$
Then $(x-1)(y-2)=2.$ Try to continue from here. good luck!