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Let-f-x-x-2013-2-a-Find-the-remainder-when-f-x-is-divided-by-x-1-done-b-Show-that-when-7-2013-2-is-divided-by-8-the-remainder-is-1-please-help-

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Question Number 9144 by Mathcheung last updated on 21/Nov/16
Let f(x)=x^(2013) +2 (a).Find the remainder when f(x) is divided by x+1.(done) (b). Show that when 7^(2013) +2 is divided by 8, the remainder is 1. (please help!)
$${Let}\:{f}\left({x}\right)={x}^{\mathrm{2013}} +\mathrm{2} \\ $$$$\left({a}\right).{Find}\:{the}\:{remainder}\:{when}\:{f}\left({x}\right)\:{is}\:{divided}\:{by}\:{x}+\mathrm{1}.\left({done}\right) \\ $$$$\left({b}\right).\:{Show}\:{that}\:{when}\:\mathrm{7}^{\mathrm{2013}} +\mathrm{2}\:{is}\:{divided}\:{by}\:\mathrm{8},\:{the}\:{remainder} \\ $$$$\:\:\:\:\:\:\:\:\:\:{is}\:\mathrm{1}.\:\:\:\:\left({please}\:{help}!\right) \\ $$$$ \\ $$
Answered by RasheedSoomro last updated on 21/Nov/16
7^2 ≡1(mod 8) (7^2 )^(1006) ≡(1)^(1006) (mod 8) 7^(2012) ≡1(mod 8) 7^(2012) ×7≡1×7(mod 8) 7^(2013) +2≡7+2(mod 8) 7^(2013) +2≡9(mod 8) 7^(2013) +2≡1(mod 8)
$$\mathrm{7}^{\mathrm{2}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\left(\mathrm{7}^{\mathrm{2}} \right)^{\mathrm{1006}} \equiv\left(\mathrm{1}\right)^{\mathrm{1006}} \left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\mathrm{7}^{\mathrm{2012}} \equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\mathrm{7}^{\mathrm{2012}} ×\mathrm{7}\equiv\mathrm{1}×\mathrm{7}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\mathrm{7}^{\mathrm{2013}} +\mathrm{2}\equiv\mathrm{7}+\mathrm{2}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\mathrm{7}^{\mathrm{2013}} +\mathrm{2}\equiv\mathrm{9}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$$$\mathrm{7}^{\mathrm{2013}} +\mathrm{2}\equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$
Commented by Mathcheung last updated on 21/Nov/16
Thank you!
$${Thank}\:{you}! \\ $$

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