The number of pairs of integers whose sums are equal to their products is

Let the two integers be $$x$$ and $$y$$

Given, $$x+y=xy$$

or, $$y=xy-x$$

or,$$y=x(y-1)$$

or, $$x=\dfrac{y}{y-1}=1+\dfrac{1}{y-1}$$

So, for $$x$$ to be an integer, the denominator $$y-1$$ can take only two possible values 1 and -1. In any other case, the value will be a fraction not an integer.

so, $$y-1=1$$,or, $$y=2$$ and so, $$x=2$$

$$y-1=-1$$,or, $$y=0$$ and so, $$x=0$$

So, the integer pairs are (0,0) and (2,2)

So, 2 pairs of integers are possible.