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Algebra

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 Joined: 29 Jun 2012 Post: 13
22 Aug 2012 21:54:25 IST
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divisibility
Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Algebra

p = 2008^2007 - 2008 and q =2008^2+2009 find the remainder when p divided by q

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Joined: 19 Jan 2008
Posts: 1070
24 Aug 2012 11:37:45 IST
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$\hspace{-16} Here \bf{p=2008^{2007} - 2008} and \bf{q=2008^{2}+2008+1}\\\\\\ Let \bf{x=2008}.Then\\\\\\ \bf{p=x^{2007}-x=\left(x^{3}\right)^{669}-1+(1-x)}\\\\\\ \bf{=\left(x^3-1\right).Q(x)+(1-x)}\\\\\\ \bf{=(x-1)(x^2+x+1).Q(x)-(x^2+x+1)+(x^2+2)}\\\\\\ Using \bf{*x^n-1=(x-1).Q(x)\forall n\in \mathbb{Z^{+}}(odd\;+ve\; Integer)}\\\\\\ and \bf{q=x^2+x+1}\\\\\\ So Remainder\bf{\left(\frac{p}{q}\right)}= Remainder\bf{\left(\frac{x^{2007}-x}{x^2+x+1}\right)}\\\\\\ \bf{=\frac{(x-1)(x^2+x+1).Q(x)-(x^2+x+1)+(x^2+2)}{x^2+x+1}}\\\\\\ \bf{=x^2+2}\\\\\\ So Remainder is \bf{=x^2+2=(2008)^2+2}$

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