Find the integral solutions to the equation $[x] [y] = x + y$ . Show that all the non-integral solutions lie on exactly two lines. Determine these lines. Here[.] denotes greatest integer function.
$x = I_{1} + f_{1} y = I_{2} + f_{2}$
then $I_{1} I_{2} = I_{1} + I_{2} + f_{1} + f_{2}$

Question

Find the integral solutions to the equation $[x] [y] = x + y$ . Show that all the non-integral solutions lie on exactly two lines. Determine these lines. Here[.] denotes greatest integer function.

$x = I_{1} + f_{1} y = I_{2} + f_{2}$

then $I_{1} I_{2} = I_{1} + I_{2} + f_{1} + f_{2}$

Solution:

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