Question Number 199864 by cortano12 last updated on 10/Nov/23 $$\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{2019}} {\sum}}\mathrm{n}^{\mathrm{4}} \:\mathrm{when} \\ $$$$\:\:\mathrm{divide}\:\mathrm{by}\:\mathrm{53}\: \\ $$ Answered by AST last updated on 10/Nov/23 $$\underset{{n}=\mathrm{1}}…
Question Number 199311 by necx122 last updated on 01/Nov/23 $${Find}\:{the}\:{number}\:{of}\:{positive}\:{integers} \\ $$$${that}\:{are}\:{factors}\:{of}\:\mathrm{3}^{\mathrm{19}} .\mathrm{7}^{\mathrm{12}} .\mathrm{10}^{\mathrm{25}} \:{and}\:{are} \\ $$$${also}\:{multiples}\:{of}\:\mathrm{3}^{\mathrm{15}} .\mathrm{7}^{\mathrm{10}} .\mathrm{10}^{\mathrm{19}} \\ $$ Answered by AST last…
Question Number 199331 by necx122 last updated on 01/Nov/23 $${What}\:{is}\:{the}\:{remainder}\:{when} \\ $$$$\mathrm{1}^{\mathrm{1}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{3}} +……+\mathrm{2023}^{\mathrm{2023}} \:{is}\:{divided}\:{by}\:\mathrm{7} \\ $$ Answered by AST last updated on 01/Nov/23…
Question Number 199011 by necx122 last updated on 26/Oct/23 $${Sum}\:{of}\:{two}\:{irrational}\:{numbers}\:{is}\:\mathrm{1} \\ $$$${less}\:{than}\:{their}\:{product},\:{and}\:\mathrm{8}\:{less}\:{than} \\ $$$${their}\:{sum}\:{of}\:{squares}.\:{Find}\:{the}\:{larger} \\ $$$${of}\:{the}\:{two}\:{numbers}. \\ $$ Commented by nikif99 last updated on 27/Oct/23…
Question Number 198684 by cortano12 last updated on 23/Oct/23 $$\:\cancel{\zeta} \\ $$ Answered by Red1ight last updated on 23/Oct/23 $$\mathrm{24},\mathrm{88} \\ $$$${A}=\mathrm{24}×\mathrm{88}=\mathrm{2112} \\ $$$${P}=\mathrm{24}+\mathrm{88}=\mathrm{112} \\…
Question Number 198418 by cortano12 last updated on 19/Oct/23 $$\:\mathrm{20}^{\mathrm{22}} −\mathrm{1}\:=\:…\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$$$ \\ $$ Answered by BaliramKumar last updated on 19/Oct/23 $$ \\ $$$$\frac{\mathrm{20}^{\mathrm{22}}…
Question Number 198400 by cortano12 last updated on 19/Oct/23 $$\:\:\mathrm{20}^{\mathrm{11}} −\mathrm{1}\:=\:…\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$ Answered by MM42 last updated on 19/Oct/23 $$\mathrm{20}^{\mathrm{11}} −\mathrm{1}\overset{\mathrm{1000}} {\equiv}−\mathrm{1}\overset{\mathrm{1000}} {\equiv}\mathrm{999} \\…
Question Number 197752 by cortano12 last updated on 27/Sep/23 $$\:\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{m} \\ $$$$\:\mathrm{such}\:\mathrm{that}\:\mathrm{m}^{\mathrm{19}} =\:\mathrm{1800}\:\left(\mathrm{mod}\:\mathrm{2029}\right) \\ $$ Answered by AST last updated on 27/Sep/23 $${m}^{\mathrm{2028}} =\left({m}^{\mathrm{19}} \right)^{\mathrm{107}}…
Question Number 197461 by Erico last updated on 18/Sep/23 $$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\bullet\underset{\:\mathrm{0}} {\int}^{\:\mathrm{x}} \frac{\mathrm{lnt}}{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\mathrm{dt}=\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{arctan}\left(\mathrm{xtan}\theta\right)\mathrm{d}\theta \\ $$$$\bullet\:\:\underset{\:\frac{\mathrm{1}}{\mathrm{x}}} {\int}^{\:\mathrm{x}} \frac{\mathrm{lnt}}{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\mathrm{arctant}\:\mathrm{dt}=\frac{\pi}{\mathrm{8}}\underset{\:\mathrm{0}} {\int}^{\:\pi} \mathrm{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{sint}\right)\mathrm{dt} \\…
Question Number 196676 by Rasheed.Sindhi last updated on 29/Aug/23 $${If}\:\overline {{ab}}\:\centerdot\:\overline {{cd}}=\mathrm{899}\:,\:{find}\:\Omega=\:\overline {{abcd}}\:+\:\overline {{cdab}} \\ $$ Commented by Rasheed.Sindhi last updated on 29/Aug/23 $${Q}#\mathrm{196659} \\…