Find-all-integers-x-and-y-such-that-xy-x-y-is-also-integer-

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 \dots t h e n y = 0, - 1, - 2, - 3 \dots

2) x + y = 1 y = 1, 2, 3, 4 \dots t h e n x = 0, - 1, - 2 \dots

s i r i a m t r y i n g t o f i n d m o r e \dots

x + y = - 1 x = 0, 1, 2, 3 \dots y = - 1, - 2, - 3 \dots

x + y = - 1 y = 0, 1, 2, 3 \dots x = - 1, - 2, - 3, \dots

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 \dots

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 \dots 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 \dots

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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