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Author: Tinku Tara

Question-201949

Question Number 201949 by sonukgindia last updated on 16/Dec/23 Answered by Frix last updated on 17/Dec/23 $${n}^{{k}} \equiv{n}\mathrm{mod2024};\:{n}\in\left\{\mathrm{529},\:\mathrm{737},\:\mathrm{760},\:\mathrm{1265},\:\mathrm{1288},\:\mathrm{1496}\right\} \\ $$$$\Rightarrow \\ $$$$\Sigma{n}^{{n}} \equiv\mathrm{3mod2024} \\ $$…

Question-201947

Question Number 201947 by Mingma last updated on 16/Dec/23 Answered by Frix last updated on 18/Dec/23 $$\mathrm{sin}\:\alpha\:\mathrm{sin}\:\beta\:\mathrm{cos}\:\left(\pi−\alpha−\beta\right)\:+ \\ $$$$\:\:\:\:\:+\mathrm{sin}\:\beta\:\mathrm{sin}\:\left(\pi−\alpha−\beta\right)\:\mathrm{cos}\:\alpha\:+ \\ $$$$\:\:\:\:\:+\mathrm{sin}\:\left(\pi−\alpha−\beta\right)\:\mathrm{sin}\:\alpha\:\mathrm{cos}\:\beta\:= \\ $$$$=\frac{\mathrm{3}−\left(\mathrm{cos}\:\mathrm{2}\alpha\:+\mathrm{cos}\:\mathrm{2}\beta\:+\mathrm{cos}\:\mathrm{2}\left(\alpha+\beta\right)\right.}{\mathrm{4}}= \\ $$$$\:\:\:\:\:\left[\mathrm{Let}\:\alpha={u}−{v}\wedge\beta={u}+{v}\right]…

Question-201956

Question Number 201956 by sonukgindia last updated on 16/Dec/23 Commented by mahdipoor last updated on 16/Dec/23 $$\mathrm{5}^{\mathrm{2}^{\mathrm{117}} } −\mathrm{1}=\left(\mathrm{5}^{\mathrm{2}^{\mathrm{116}} } −\mathrm{1}\right)\left(\mathrm{5}^{\mathrm{2}^{\mathrm{116}} } +\mathrm{1}\right)= \\ $$$$\left(\mathrm{5}^{\mathrm{2}^{\mathrm{115}}…