x, y are positive integer such that, x^3 +y^3 +xy=911. (x,y)=?


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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

${T h a t}^{'} s a s y m m e t r i c F u n c t i o n$ ${(x + y)}^{3} - 3 x y (x + y) + x y = 911$ $a^{3} - 3 b a + b = 911$ $b = \frac{a^{3} - 911}{3 a - 1}$ $T r y i n g a = 10, 11, . . .$ $b = \frac{15^{3} - 911}{3 (15) - 1} = 56$ $x + y = 15, x y = 56$ $∴ (x, y) = (8, 7), (7, 8)$
	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

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