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