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What-is-the-remainder-when-13-5-14-5-15-5-16-5-is-divided-by-29-




Question Number 9021 by tawakalitu last updated on 14/Nov/16
What is the remainder when   (13^5  + 14^5  + 15^5  + 16^5 ) is divided by 29 ?
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\: \\ $$$$\left(\mathrm{13}^{\mathrm{5}} \:+\:\mathrm{14}^{\mathrm{5}} \:+\:\mathrm{15}^{\mathrm{5}} \:+\:\mathrm{16}^{\mathrm{5}} \right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{29}\:?\: \\ $$
Answered by aydnmustafa1976 last updated on 14/Nov/16
13^5 +16^5 +14^5 +15^5 =13^5 +(−13)^5 +14^5 +(−14)^5 =13^5 +−13^5 +14^5 +−14^5 =0
$$\mathrm{13}^{\mathrm{5}} +\mathrm{16}^{\mathrm{5}} +\mathrm{14}^{\mathrm{5}} +\mathrm{15}^{\mathrm{5}} =\mathrm{13}^{\mathrm{5}} +\left(−\mathrm{13}\right)^{\mathrm{5}} +\mathrm{14}^{\mathrm{5}} +\left(−\mathrm{14}\right)^{\mathrm{5}} =\mathrm{13}^{\mathrm{5}} +−\mathrm{13}^{\mathrm{5}} +\mathrm{14}^{\mathrm{5}} +−\mathrm{14}^{\mathrm{5}} =\mathrm{0} \\ $$
Commented by tawakalitu last updated on 14/Nov/16
thanks sir.
$$\mathrm{thanks}\:\mathrm{sir}. \\ $$

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