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[Solved] Roots and equations finding solutions

 
Linear Equations
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June 14, 2021 6:58 pm
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(@armor)
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Find all positive integer solutions (x,y) of the equation  x\sqrt{y}+y\sqrt{x}-\sqrt{2011x}-\sqrt{2011y}+\sqrt{2011xy}=2011.

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June 14, 2021 6:59 pm
Topic starter
Staff 
(@armor)
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317 Posts
156 153 0

The equation is equivalent to 

\small \sqrt{xy}\sqrt{x}+\sqrt{xy}\sqrt{y}-\sqrt{2011x}-\sqrt{2011y}+\sqrt{2011xy}-(\sqrt{2011})^2=0\Leftrightarrow \sqrt{xy}(\sqrt{x}+\sqrt{y})-\sqrt{2011}(\sqrt{x}+\sqrt{y})+\sqrt{2011}\sqrt{xy}-(\sqrt{2011})^2=0\Leftrightarrow (\sqrt{xy}-\sqrt{2011})(\sqrt{x}+\sqrt{y}+\sqrt{2011})=0.

Since \small \sqrt{x}+\sqrt{y}+\sqrt{2011}>0,  then  \small \sqrt{xy}-\sqrt{2011}=0\Rightarrow xy=2011.

Since 2011 is a prime, then x=1, y=2011 or y=1, x=2011. Hence the original equation has two positive integer solutions (1,2011), (2011,1).

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