Question Number 29036 by abdo imad last updated on 03/Feb/18 $${p}=\mathrm{2}{m}+\mathrm{1}\:{is}\:{a}\:{prime}\:{number}\:{prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\left({p}−\mathrm{1}\right)!\equiv\:−\mathrm{1}\left[{p}\right] \\ $$$$\left.\mathrm{2}\right)\:\left({m}!\right)^{\mathrm{2}} \equiv\:\left(−\mathrm{1}\right)^{{m}+\mathrm{1}} \:\left[{p}\right] \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 94553 by john santu last updated on 19/May/20 $$\mathrm{solve}\:\frac{\mid{x}+\mathrm{2}\mid−{x}}{{x}}\:<\:\mathrm{2}\: \\ $$ Answered by i jagooll last updated on 19/May/20 Terms of Service Privacy…
Question Number 160052 by mr W last updated on 24/Nov/21 $${find}\:\Phi\left({k}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{{k}} }{{n}!}\:{with}\:{k}\geqslant\mathrm{1}. \\ $$ Answered by Tokugami last updated on 24/Nov/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty}…
Question Number 28932 by Rasheed.Sindhi last updated on 01/Feb/18 $$\mathrm{Determine}\:\mathrm{the}\:\mathrm{least}\:\mathrm{number}\:\mathrm{of}\:\mathrm{4}\:\mathrm{digits}, \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{perfect}\:\mathrm{square}. \\ $$$$\mathrm{Method}\:\mathrm{of}\:\mathrm{finding}\:\mathrm{is}\:\boldsymbol{\mathrm{required}}. \\ $$ Answered by mrW2 last updated on 01/Feb/18 $${x}={n}^{\mathrm{2}} \geqslant\mathrm{1000}…
Question Number 159891 by tounghoungko last updated on 22/Nov/21 $$\:\:\:\:{S}=\underset{{k}=\mathrm{1}} {\overset{\mathrm{2002}\:} {\sum}}\sqrt{\frac{{k}^{\mathrm{2}} +\mathrm{1}}{{k}^{\mathrm{2}} }+\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }}\:=? \\ $$ Answered by chhaythean last updated on 22/Nov/21 $$\mathrm{S}=\underset{\mathrm{k}=\mathrm{1}}…
Question Number 28805 by NECx last updated on 30/Jan/18 $${In}\:{a}\:{competition},\:{a}\:{school}\:{awarded} \\ $$$${medals}\:{in}\:{different}\:{categories}. \\ $$$$\mathrm{36}\:{medals}\:{in}\:{dance},\mathrm{12}\:{in}\:{dramatics} \\ $$$${and}\:\mathrm{18}\:{medals}\:{in}\:{music}.{If}\:{these} \\ $$$${medals}\:{went}\:{to}\:{total}\:\mathrm{45},{and}\:{only} \\ $$$$\mathrm{4}\:{persons}\:{got}\:{medals}\:{in}\:{all}\:{three} \\ $$$${catogories}.{Using}\:{set}\:{notations}, \\ $$$${how}\:{many}\:{received}\:{in}\:{exactly} \\…
Question Number 94298 by peter frank last updated on 17/May/20 Answered by Ar Brandon last updated on 18/May/20 $$\mathrm{log}_{\mathrm{2}} \left(\mathrm{1}×\mathrm{2}×\mathrm{3}×…×\mathrm{n}\right)=\mathrm{1994} \\ $$$$\mathrm{log}_{\mathrm{2}} \left(\mathrm{n}!\right)=\mathrm{1994}\Rightarrow\mathrm{n}!=\mathrm{2}^{\mathrm{1994}} \\ $$$$\mathrm{n}\approx\mathrm{295}…
Question Number 94297 by peter frank last updated on 17/May/20 Commented by mathmax by abdo last updated on 18/May/20 $${we}\:{know}\:{that}\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}} =\mathrm{0}\:\Rightarrow{x}={y}={z}\:\:\:\left({for}\:{x},{y}\:,{z}\:{reals}\right) \\ $$$${so}\:\left({e}\right)\:\Rightarrow{cosx}\:={cos}\left(\mathrm{2}{x}\right)={cos}\left(\mathrm{3}{x}\right)=\mathrm{0}…
Question Number 28743 by Cheyboy last updated on 29/Jan/18 $${find}\:{the}\:{next}\:\mathrm{4}\:{term}\:{and}\:{the} \\ $$$${n}^{{th}} \:{term} \\ $$$$\mathrm{1},\mathrm{2},\mathrm{5},\mathrm{26}……. \\ $$ Commented by Tinkutara last updated on 29/Jan/18 $${a}_{{n}}…
Question Number 159750 by greg_ed last updated on 20/Nov/21 $$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\:\sqrt{\mathrm{689}×\mathrm{690}×\mathrm{691}+\mathrm{1}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{naturel}}\:\boldsymbol{\mathrm{number}}. \\ $$ Commented by Rasheed.Sindhi last updated on 20/Nov/21 $${Pl}\:{check}\:{the}\:{question},\:\sqrt{\mathrm{689}×\mathrm{690}×\mathrm{691}+\mathrm{1}}\: \\ $$$${is}\:{not}\:{a}\:{natural}\:{number}. \\…