## Junior Divisional 2013/3

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### Junior Divisional 2013/3

Find out the greatest integer $n$ for which $n^3+500$ will be divisible by $n+10$.

### Re: Faridpur, 12th bdmo , j19

Please,use $Latex$ in writing equations so that the equations look beautiful.BTW,here is the solution:

$n^{3}+500=n^{3}+10^{3}-500$

$=(n+10)(n^{2}-10n+100)-500$

$(n+10)$ divides $(n+10)(n^{2}-10n+100)$.So,$(n+10)$ must divide $500$.The greatest value of $(n+10)$ which divides $500$ is $500$.$\therefore$ the greatest value of $n$ is $490$

$n^{3}+500=n^{3}+10^{3}-500$

$=(n+10)(n^{2}-10n+100)-500$

$(n+10)$ divides $(n+10)(n^{2}-10n+100)$.So,$(n+10)$ must divide $500$.The greatest value of $(n+10)$ which divides $500$ is $500$.$\therefore$ the greatest value of $n$ is $490$

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