<s>[INST] Nonnegative Integer solutions of $x+y-xy=0$ I would like to see other methods, besides the one I use here to find all the nonnegative integer solutions of an equation like $$x+y-xy=0$$.
This is the one I used:

First we note that for $x=1$ we get $1+y-y=0$, but $1+y-y=1$ so $x\neq 1$.
  Then as $x\neq 1$ we have $y=\frac{x}{x-1}=1+\frac{1}{x-1}$. Furthermore $y(0)=0$ and $y(2)=2$ so $x=y=0$ and $x=y=2$ are solutions. Then if $x>2$, $x-1>1>0$ so as $x-1>1$ we have $\frac{1}{x-1}<1$ and as $x-1>0$ we have $\frac{1}{x-1}>0$ so $0<\frac{1}{x-1}<1$ for $x>2$, and so $1<1+\frac{1}{x-1}<2$ for $x>2$. As $y=1+\frac{1}{x-1}$, we get that for $x>2$, $1<y<2$. As there is no integer between $1$ and $2$, for $x>2$, $y$ can't be a integer. So the only nonnegative solutions of the equations are $x=y=0$ and $x=y=2$.

How else can I prove this, using less theory (For example, I think it can be proven using only Peano and a little more)?
 [/INST] If
$x+y-xy = 0$,
then
$1
=xy-x-y+1
=(x-1)(y-1)
$.
Therefore
$x-1$ and $y-1$
must be integers
that divide $1$.
The only solutions are
$x=y=2$
and
$x=y=0$.
</s>