Question Number 166475 by Rasheed.Sindhi last updated on 20/Feb/22 $$\mathcal{F}{ind}\:{out}\:{n}\in\mathbb{N} \\ $$$$\:{such}\:{that} \\ $$$${n}^{\mathrm{2}} +{n}\:{is}\:{divisible}\:{by}\:\mathrm{30}. \\ $$ Commented by mr W last updated on 20/Feb/22…
Question Number 100888 by bramlex last updated on 29/Jun/20 $${Find}\:{all}\:{pairs}\:\left({k},{n}\right)\:{of}\:{positive}\: \\ $$$${integer}\:{for}\:{which}\:\mathrm{7}^{{k}} −\mathrm{3}^{{n}} \:{divides} \\ $$$${k}^{\mathrm{4}} +{n}^{\mathrm{2}} \:. \\ $$ Terms of Service Privacy Policy…
Question Number 100733 by john santu last updated on 28/Jun/20 $$\mathrm{A}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{as}\:\mathrm{4334}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{palindrome}\:\mathrm{if}\:\mathrm{it}\:\mathrm{reads}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{forwards}\:\mathrm{or}\:\mathrm{backwards}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{only}\:\mathrm{prime}\:\mathrm{palindrome}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{even}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}?\: \\ $$ Commented by Rasheed.Sindhi last…
Question Number 100677 by bobhans last updated on 28/Jun/20 $$\mathrm{for}\:\mathrm{m},\mathrm{n}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{m}\:>\:\mathrm{n}\: \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{lcd}\left(\mathrm{m},\mathrm{n}\right)\:+\:\mathrm{lcd}\left(\mathrm{m}+\mathrm{1},\mathrm{n}+\mathrm{1}\right)\:>\:\frac{\mathrm{2mn}}{\:\sqrt{\mathrm{m}−\mathrm{n}}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 35072 by Rasheed.Sindhi last updated on 15/May/18 $$\mathrm{An}\:\mathrm{n}-\mathrm{digit}\:\mathrm{decimal}\:\mathrm{number}\:\mathrm{has}\:\mathrm{been} \\ $$$$\mathrm{conveerted}\:\mathrm{into}\:\mathrm{octal}\:\mathrm{number}.\mathrm{Say}\:\mathrm{it} \\ $$$$\mathrm{has}\:\mathrm{m}\:\mathrm{digits}.\mathrm{What}\:\mathrm{are}\:\mathrm{possible}\:\:\mathrm{minimum}\: \\ $$$$\mathrm{and}\:\mathrm{maximum}\:\mathrm{values}\:\mathrm{of}\:\mathrm{m}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of} \\ $$$$\mathrm{n}? \\ $$ Commented by candre last updated…
Question Number 35071 by Rasheed.Sindhi last updated on 15/May/18 $$\mathrm{A}\:\mathrm{20}-\mathrm{digit}\:\mathrm{decimal}\:\mathrm{number}\:\mathrm{has}\:\mathrm{been} \\ $$$$\mathrm{converted}\:\mathrm{into}\:\mathrm{octal}\:\mathrm{system}.\mathrm{Say}\:\mathrm{it}\:\mathrm{has}\: \\ $$$$\mathrm{n}\:\mathrm{digits}.\:\mathrm{What}\:\mathrm{can}\:\mathrm{be}\:\mathrm{minimum}\:\mathrm{and} \\ $$$$\mathrm{maximum}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}? \\ $$$$ \\ $$ Answered by candre last updated…
Question Number 100583 by Rio Michael last updated on 27/Jun/20 $$\:\mathrm{A}\:\mathrm{transformation}\:{f}\:\mathrm{on}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{plane} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{z}'\:=\:\left(\mathrm{1}\:+{i}\right){z}\:−\mathrm{3}\:+\:\mathrm{4}{i} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{simultitude}\:\mathrm{with}\:\mathrm{radius}\:{r}\:\mathrm{and}\:\mathrm{centre} \\ $$$$\Omega\:\mathrm{to}\:\mathrm{be}\:\mathrm{determined}. \\ $$$$\mathrm{Determine}\:\mathrm{to}\:\mathrm{the}\:\mathrm{invariant}\:\mathrm{point}\:\mathrm{under}\:{f}. \\ $$ Terms of Service Privacy…
Question Number 35004 by Rasheed.Sindhi last updated on 14/May/18 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{41}\:\mathrm{divides}\:\mathrm{2}^{\mathrm{20}} −\mathrm{1} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 14/May/18 $${let}\:{us}\:{solve}\:{analytically}… \\ $$$$\mathrm{2}^{\mathrm{20}} −\mathrm{1}=\left(\mathrm{2}^{\mathrm{5}} \right)^{\mathrm{4}}…
Question Number 35005 by Rasheed.Sindhi last updated on 14/May/18 $$\mathrm{Find}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\:\:\:\:\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+…+\mathrm{99}!+\mathrm{100}! \\ $$$$\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{12}. \\ $$ Answered by rahul 19 last updated on 14/May/18 $$\mathrm{9}\:.…
Question Number 100179 by bobhans last updated on 25/Jun/20 Answered by john santu last updated on 25/Jun/20 $$\mathrm{Tools}\:\sqrt{\mathrm{A}+\sqrt{\mathrm{B}}}\:=\:\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{A}+\sqrt{\mathrm{A}^{\mathrm{2}} −\mathrm{B}}\right)}\:+\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{A}−\sqrt{\mathrm{A}^{\mathrm{2}} −\mathrm{B}}\right)} \\ $$$$\Rightarrow\sqrt{\mathrm{m}+\sqrt{\mathrm{m}^{\mathrm{2}} −\mathrm{1}}}\:=\:\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{m}+\mathrm{1}\right)}+\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{m}−\mathrm{1}\right)} \\ $$$$\underset{\mathrm{m}\:=\:\mathrm{1}\:}…