Find all integers x and y such that ((xy)/(x+y)) is also integer.


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Question Number 53144 by mr W last updated on 18/Jan/19

$F i n d a l l i n t e g e r s x a n d y s u c h t h a t$ $\frac{x y}{x + y} i s a l s o i n t e g e r .$
Commented by mr W last updated on 19/Jan/19

$x = (i + 1) j$ $y = i (i + 1) j$ $i, j \in Z$
	Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jan/19

	$x = 0 y = 1 \frac{x y}{x + y} = 0 a n d x = 1 y = 0 \frac{x y}{x + y} = 0$ $\frac{x y}{x + y} i s i n t e g e r w h e n$ $1) x + y = 1 x = 1, 2, 3, 4 . . . t h e n y = 0, - 1, - 2, - 3 . . .$ $2) x + y = 1 y = 1, 2, 3, 4 . . . t h e n x = 0, - 1, - 2 . . .$ $s i r i a m t r y i n g t o f i n d m o r e . . .$ $x + y = - 1 x = 0, 1, 2, 3 . . . y = - 1, - 2, - 3 . . .$ $x + y = - 1 y = 0, 1, 2, 3 . . . x = - 1, - 2, - 3, . . .$
	Commented by mr W last updated on 18/Jan/19

	$t h a n k s f o r t r y i n g!$ $c a n y o u g i v e a g e n e r a l s o l u t i o n f o r$ $a l l s o l u t i o n s ?$
	Commented by tanmay.chaudhury50@gmail.com last updated on 18/Jan/19

	$s i r i a m t r y i n g . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 18/Jan/19

	$i t h i n k g e n e r a l s o l u t i o n r e l a t e d t o g r e a t e s t i n t e g e r$ $f u n c t i o n . . . b e t t e r y o u p o s t g e n e t a l s o l u t i o n s i r . . .$ $a n d$
	Answered by ajfour last updated on 18/Jan/19

	$\frac{1}{N} = \frac{1}{x} + \frac{1}{y} = \frac{x + y}{x y}$ $\Rightarrow z^{2} - z + N = 0$ $x, y = \frac{1 \pm \sqrt{1 - 4 N}}{2}$ $\Rightarrow 1 - 4 N = {(2 k - 1)}^{2}$ $\Rightarrow - 4 N = 4 k^{2} - 4 k$ $o r N = k - k^{2}$ $x, y = \frac{1 \pm ∣ 2 k - 1 ∣}{2} .$
	Commented by ajfour last updated on 18/Jan/19

	$w i l l t h i s d o S i r ?$
	Commented by mr W last updated on 18/Jan/19

	$i t d e s c r i b s o n l y s o m e s o l u t i o n s, i . e .$ $t h o s e w i t h x + y = 1$ $b u t n o t a l l s o l u t i o n s, s o i t i s n o t$ $a g e n e r a l s o l u t i o n s i r, e . g .$ $f o l l o w i n g s o l u t i o n s a r e c o r r e c t, b u t t h e y$ $c a n n o t b e o b t a i n e d f r o m y o u r f o r m u l a :$ $x = 2, y = 2$ $x = 3, y = 6$
	Commented by ajfour last updated on 18/Jan/19

	$o k a y s i r, i s u s p e c t e d s o t o o, s h a l l$ $t r y a g a i n .$
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