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The number of pairs of integers whose sums are equal to their products is
Correct Answer: 2
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.
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