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Question Number 159863 by HongKing last updated on 21/Nov/21

$$\mathrm{Dertermine}\mathrm{all}\mathrm{pairs}(\mathbf{x};\mathbf{y})\mathrm{of}\mathrm{integers}$$$$\mathrm{such}\mathrm{that}$$$$2010\mathrm{x}-\mathrm{xy}+2012\mathrm{y}+1=0$$

Answered by mr W last updated on 21/Nov/21

$$(x-2012)y=1+2010x$$$$y=\frac{1+2010\times 2012+2010(x-2012)}{x-2012}$$$$y=2010+\frac{{2011}^2}{x-2012}$$$$x-2012=\pm 1$$$$\Rightarrow x=2011,2013$$$$\Rightarrow y=-4042111,4046131$$$$x-2012=\pm 2011$$$$\Rightarrow x=1,4023$$$$\Rightarrow y=-1,4021$$$$x-2012=\pm 4044121$$$$\Rightarrow x=-4042109,4046133$$$$\Rightarrow y=2009,2011$$$$allpairs(x,y):$$$$(2011,-4042111)$$$$(2013,4046131)$$$$(1,-1)$$$$(4023,4021)$$$$(-4042109,2009)$$$$(4046133,2011)$$

Commented by Rasheed.Sindhi last updated on 22/Nov/21

$${\mathbf{e}}^{\mathbf{x}}\mathrm{cellent}\mathrm{sir}!$$

Commented by HongKing last updated on 22/Nov/21

$$\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}\mathrm{my}\mathrm{dear}\mathrm{Ser}\mathrm{cool}$$

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