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} \\…
Question Number 196672 by Rasheed.Sindhi last updated on 29/Aug/23 $${Find}\:{all}\:\Omega\overline {={abcdef}\:\:},\:{such}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{abcdef}={abc}+{def} \\ $$ Commented by Rasheed.Sindhi last updated on 29/Aug/23 $${Q}#\mathrm{196659} \\ $$…
Question Number 196053 by mathlove last updated on 17/Aug/23 $${faind}\:{n}\:{terme} \\ $$$$\mathrm{4},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{9}},−\mathrm{2},…… \\ $$ Answered by Frix last updated on 17/Aug/23 $$\frac{\mathrm{16}}{\mathrm{4}},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{9}},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{14}},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{19}},… \\ $$$$\frac{\mathrm{16}}{\mathrm{4}},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{9}},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{16}},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{25}},… \\…
Question Number 195443 by MM42 last updated on 02/Aug/23 $$\mathrm{10}^{\mathrm{10}} +\mathrm{10}^{\mathrm{10}^{\mathrm{2}} } +\mathrm{10}^{\mathrm{10}^{\mathrm{3}} } +…+\mathrm{10}^{\mathrm{10}^{\mathrm{10}} } \:\:\overset{\mathrm{7}} {\equiv}\:\:? \\ $$ Answered by BaliramKumar last updated…
Question Number 194689 by cortano12 last updated on 13/Jul/23 Answered by MM42 last updated on 13/Jul/23 $${f}_{{i}} :\mathrm{1},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8},\mathrm{13},\mathrm{21},\mathrm{34},\mathrm{55},\mathrm{89},… \\ $$$$\Rightarrow\forall\:{n}=\mathrm{5}{k}\:\:;\:\mathrm{5}\mid{f}_{{n}} \:\:\:!!?? \\ $$$$\Rightarrow{if}\:\:{n}<\mathrm{10000}\:\rightarrow\:{ans}=\mathrm{4999}\:\checkmark \\ $$$${if}\:\:{f}_{{n}}…
Question Number 193485 by Socracious last updated on 15/Jun/23 $$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} −\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{divisible}}\:\boldsymbol{\mathrm{by}} \\ $$$$\:\:\:\:\:\:\:\mathrm{3}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{n}}. \\ $$ Answered by MM42 last updated on 15/Jun/23…