Question Number 88050 by Sahil vampire last updated on 08/Apr/20 Commented by mr W last updated on 08/Apr/20 $${i}\:{found}\:{there}\:{is}\:{only}\:{one}\:{such}\:{number}: \\ $$$$\mathrm{588}\:\mathrm{2353} \\ $$$$\mathrm{588}^{\mathrm{2}} +\mathrm{2353}^{\mathrm{2}} =\mathrm{588}\:\mathrm{2353}…
Question Number 88000 by jdmath last updated on 07/Apr/20 $${Is}\:{there}\:{a}\:{formula}\:{to}\:{calculate}\: \\ $$$$ \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{i}^{\mathrm{2}} } \\ $$$${interms}\:{of}\:{n}..? \\ $$ Commented by abdomathmax last…
Question Number 153458 by liberty last updated on 07/Sep/21 $${Given}\:{a}\:{set}\:{consisting}\:{of}\:\mathrm{22}\:{integer} \\ $$$$\:{A}=\left\{\pm{a}_{\mathrm{1}} ,\pm{a}_{\mathrm{2}} ,…,\pm{a}_{\mathrm{11}} \right\}.\:{Show}\:{that} \\ $$$${exist}\:{subset}\:{of}\:{S}\:{with}\:{properties} \\ $$$$\left(\mathrm{1}\right)\:{for}\:{every}\:{i}=\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{11}\: \\ $$$$\:{have}\:{least}\:{one}\:{between}\:{a}_{{i}} \:{or}\:−{a}_{{i}} \\ $$$$\:{element}\:{of}\:{S} \\…
Question Number 22372 by Rasheed.Sindhi last updated on 16/Oct/17 Commented by Rasheed.Sindhi last updated on 16/Oct/17 $$\mathrm{Question}\:\mathrm{asked}\:\mathrm{by}\:\mathrm{Mr}.\:\mathrm{Tinkutara}. \\ $$$$\mathrm{reposted}\:\mathrm{for}\:\mathrm{answer}. \\ $$ Commented by Tinkutara last…
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Question Number 22080 by Tinkutara last updated on 10/Oct/17 $$\mathrm{Given}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{rational} \\ $$$$\mathrm{numbers}\:{a}\:\mathrm{and}\:{b},\:{a}\:\neq\:{b},\:\mathrm{which}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{integers}\:\mathrm{and}\:\mathrm{which}\:\mathrm{are}\:\mathrm{such}\:\mathrm{that}\:{a}\:−\:{b}, \\ $$$${a}^{\mathrm{2}} \:−\:{b}^{\mathrm{2}} ,\:{a}^{\mathrm{3}} \:−\:{b}^{\mathrm{3}} ,\:…..,\:{a}^{{n}} \:−\:{b}^{{n}} \:\mathrm{are}\:\mathrm{all} \\…
Question Number 21994 by Tinkutara last updated on 08/Oct/17 $$\mathrm{Suppose}\:{N}\:\mathrm{is}\:\mathrm{an}\:{n}-\mathrm{digit}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{the}\:{n}-\mathrm{digits}\:\mathrm{are}\:\mathrm{distinct};\:\mathrm{and} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{digits}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{5}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{n}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most}\:\mathrm{6}.\:\mathrm{Further}, \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{starting}\:\mathrm{with}\:\mathrm{any}\:\mathrm{digit}\:\mathrm{one} \\ $$$$\mathrm{can}\:\mathrm{find}\:\mathrm{a}\:\mathrm{six}-\mathrm{digit}\:\mathrm{number}\:\mathrm{with}\:\mathrm{these} \\…
Question Number 21962 by Tinkutara last updated on 07/Oct/17 $$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{16}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{property}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{numbers}\:\mathrm{of}\:{A}\:\mathrm{will} \\ $$$$\mathrm{not}\:\mathrm{exceed}\:\mathrm{1994}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are} \\ $$$$\mathrm{two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{in}\:{A}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{relatively}\:\mathrm{prime}. \\ $$ Commented by Rasheed.Sindhi…
Question Number 21784 by Tinkutara last updated on 03/Oct/17 $$\mathrm{Call}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\boldsymbol{\mathrm{good}}\:\mathrm{if}\:\mathrm{there} \\ $$$$\mathrm{are}\:{n}\:\mathrm{integers},\:\mathrm{positive}\:\mathrm{or}\:\mathrm{negative},\:\mathrm{and} \\ $$$$\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct},\:\mathrm{such}\:\mathrm{that}\:\mathrm{their} \\ $$$$\mathrm{sum}\:\mathrm{and}\:\mathrm{product}\:\mathrm{are}\:\mathrm{both}\:\mathrm{equal}\:\mathrm{to}\:{n} \\ $$$$\left(\mathrm{e}.\mathrm{g}.\:\mathrm{8}\:\mathrm{is}\:\boldsymbol{\mathrm{good}}\:\mathrm{since}\right. \\ $$$$\mathrm{8}=\mathrm{4}\centerdot\mathrm{2}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\left(−\mathrm{1}\right)\left(−\mathrm{1}\right)=\mathrm{4}+\mathrm{2}+\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$$$\left.+\mathrm{1}+\left(−\mathrm{1}\right)+\left(−\mathrm{1}\right)\right). \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{integers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{4}{k}\:+\:\mathrm{1} \\…
Question Number 21781 by Tinkutara last updated on 03/Oct/17 $$\mathrm{Which}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{10}^{\mathrm{11}} \:\mathrm{or}\:\mathrm{11}^{\mathrm{10}} ? \\ $$ Answered by Joel577 last updated on 03/Oct/17 $$\:\:\:\:\:\:\:\:\:\:\mathrm{10}^{\mathrm{11}} \:…\:\mathrm{11}^{\mathrm{10}} \\ $$$$\mathrm{11}\:\mathrm{log}\:\mathrm{10}\:…\:\mathrm{10}\:\mathrm{log}\:\mathrm{11}…