Category: Arithmetic

Question Number 18213 by chux last updated on 16/Jul/17 $$\mathrm{The}\:\mathrm{greatest}\:\mathrm{number}\:\mathrm{that}\:\mathrm{will} \\ $$$$\mathrm{divide}\:\mathrm{82}\:,\mathrm{111}\:\mathrm{and}\:\mathrm{140}\:\mathrm{leaving}\:\mathrm{the}\: \\ $$$$\mathrm{same}\:\mathrm{remainder}\:\mathrm{in}\:\mathrm{each}\:\mathrm{case}\:\mathrm{is}…….. \\ $$ Answered by Tinkutara last updated on 17/Jul/17 $$\mathrm{82}\:=\:{xa}\:+\:{r} \\…

Question Number 83629 by M±th+et£s last updated on 04/Mar/20 $${show}\:{that} \\ $$$$\underset{{n},{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}!\:{k}!}{\left({n}+{k}+\mathrm{2}\right)!}=\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$ Answered by Kamel Kamel last updated on 04/Mar/20…

Question Number 17767 by Joel577 last updated on 10/Jul/17 $$\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{30}} {\sum}}\left({n}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\: \\ $$$$\left(\mathrm{A}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{224}\right) \\ $$$$\left(\mathrm{B}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{225}\right) \\ $$$$\left(\mathrm{C}\right)\:\underset{{n}\:=\:\mathrm{1}}…

Question Number 148780 by Jonathanwaweh last updated on 31/Jul/21 Answered by Kamel last updated on 31/Jul/21 $$ \\ $$$$\Omega=\underset{{n}=\mathrm{2}} {\overset{+\infty} {\prod}}{e}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)^{{n}^{\mathrm{2}} } ={e}^{\underset{{n}=\mathrm{2}} {\overset{+\infty}…

Question Number 83242 by peter frank last updated on 29/Feb/20 Commented by john santu last updated on 29/Feb/20 $$\left(\mathrm{4b}\right)\:\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{r}\left(\mathrm{r}+\mathrm{1}\right)\:=\:\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{r}^{\mathrm{2}} \:+\:\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{n}}…