A number theory problem by Daman Deep singh

Given,

$xy-2x-2y=0$

$\Rightarrow$ $xy-2x-2y+4=4$

$\Rightarrow$ $(x-2)(y-2)=4$

Now , to find $(x,y)$ integer pairs ,

We need to find all factors of $4$ ,

Since , $\color{#20A900}{4=2^2}$ , Therefore number of positive factors of $4$ is $(2+1)=3$ .

So we have $3$ unique pairs of integers $(x,y)$ ; But we have not taken the negative factors ; Therefore, after including them ,

Our final count is $\color{#3D99F6}{\boxed {6}}$ pairs of integers $(x,y)$ .