Find integral solutions to the equation [x] [y] = x + y.Show that all integral solutions lie on two lines. Determine these lines , where [.] denotes greatest integer function.

Question
Find integral solutions to the equation [x] [y] = x + y. Show that all integral solutions lie on two lines. Determine these lines, where [.] denotes greatest integer function.

Answer
[x] [y] = x + y.
This implies x + y is an integer. Lets us assume {x} = f and {y} = f' (0<=f<1 and 0<=f'<1 and {.} denotes fractional part of x ). Also, let [x] = I and let [y] = J. So, our equation becomes :

IJ = I + J + f + f' .

So, either f + f' = 0 or f + f' = 1.

For f + f' = 0 (So, f = 0 and f' = 0) , we get :

IJ = I + J
(I-1)(J-1) = 1
(I , J) = { (0 ,0) , (2 , 2) }

For f + f' = 1 ,we get

IJ = I + J + 1
(I-1)(J-1) = 2

(I , J) = { (3 , 2) , (2 , 3) , (0 , -1) , (-1 , 0) }

So, ( using f' = 1 - f and x = I + f and y = J + f' = J + 1 -f )

(x , y) = { (3 + f , 3 - f) , (2 + f , 4 - f) , (f , -f) , (-1 + f , 1 - f) }

The solutions (x , y) lie on line segments :

x + y = 6 (2 <= x < 4)
x + y = 0 (-1 <= x < 1)

and also the point :

(x , y) = (2 , 2)