x-y-are-positive-integer-such-that-x-3-y-3-xy-911-x-y-

Question Number 211680 by RojaTaniya last updated on 16/Sep/24

x, y a r e p o s i t i v e i n t e g e r s u c h

t h a t, x^{3} + y^{3} + x y = 911 . (x, y) = ?

Commented by AlagaIbile last updated on 16/Sep/24

{T h a t}^{'} s a s y m m e t r i c F u n c t i o n

{(x + y)}^{3} - 3 x y (x + y) + x y = 911

a^{3} - 3 b a + b = 911

b = \frac{a^{3} - 911}{3 a - 1}

T r y i n g a = 10, 11, \dots

b = \frac{15^{3} - 911}{3 (15) - 1} = 56

x + y = 15, x y = 56

∴ (x, y) = (8, 7), (7, 8)

Answered by A5T last updated on 16/Sep/24

W L O G, l e t x ⩾ y \Rightarrow 911 = x^{3} + y^{3} + x y ⩾ 2 y^{3} + y^{2}

\Rightarrow y ⩽ 7 . O b s e r v e t h a t x ⩽ 9, b e c a u s e 10^{3} > 911

C h e c k i n g \Rightarrow y = 7, x = 8

\Rightarrow (x, y) = (8, 7) u p t o p e r m u t a t i o n .

Commented by RojaTaniya last updated on 16/Sep/24

S i r, p e r f e c t i d e a . T h a n k s .