Question Number 13909 by tawa tawa last updated on 25/May/17 $$\mathrm{Solve}\:\mathrm{simutaneously}. \\ $$$$\mathrm{x}\sqrt{\mathrm{x}}\:+\:\mathrm{y}\sqrt{\mathrm{y}}\:=\:\mathrm{183} \\ $$$$\mathrm{x}\sqrt{\mathrm{y}}\:+\:\mathrm{y}\sqrt{\mathrm{x}}\:=\:\mathrm{185} \\ $$ Commented by RasheedSindhi last updated on 25/May/17 $$\left(\mathrm{x}^{\mathrm{1}/\mathrm{2}}…
Question Number 144980 by henderson last updated on 01/Jul/21 $$\underline{\boldsymbol{\mathrm{exercise}}} \\ $$$$\boldsymbol{\mathrm{Let}}\:\boldsymbol{{a}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{b}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{natural}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{{a}}\:\boldsymbol{\mathrm{divides}}\:\boldsymbol{{b}},\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{naturel}}\: \\ $$$$\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{{n}}^{\boldsymbol{{a}}} −\mathrm{1}\:\boldsymbol{\mathrm{divides}}\:\boldsymbol{{n}}^{\boldsymbol{{b}}} −\mathrm{1}. \\ $$$$ \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{For}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{non}}−\boldsymbol{\mathrm{zero}}\:\boldsymbol{\mathrm{naturel}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{\mathrm{prove}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{remainder}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{euclidean}}\:\boldsymbol{\mathrm{division}}\:\boldsymbol{\mathrm{of}}\: \\…
Question Number 144928 by ArielVyny last updated on 30/Jun/21 $$\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+{z}\right){n}!} \\ $$ Answered by Dwaipayan Shikari last updated on 01/Jul/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{n}\geqslant\mathrm{0}}…
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Question Number 78979 by jagoll last updated on 22/Jan/20 $$\mathrm{the}\:\mathrm{first}\:\mathrm{n}\:\mathrm{even} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{known}.\:\mathrm{if}\:\mathrm{one}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{deleted}\:\mathrm{then}\:\mathrm{the}\:\mathrm{average} \\ $$$$\mathrm{of}\:\mathrm{remaining}\:\mathrm{numbers}\:\mathrm{is}\:\frac{\mathrm{2582}}{\mathrm{50}}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{deleted}\:\mathrm{number}. \\ $$ Commented by mr W last…
Question Number 144482 by ArielVyny last updated on 25/Jun/21 $${show}\:{that}\:\forall{n}\in\mathbb{Z}\: \\ $$$${E}\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)+{E}\left(\frac{{n}+\mathrm{2}}{\mathrm{4}}\right)+{E}\left(\frac{{n}+\mathrm{4}}{\mathrm{4}}\right)={n} \\ $$ Answered by Olaf_Thorendsen last updated on 25/Jun/21 $${x}\:=\:\mathrm{E}\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)+\mathrm{E}\left(\frac{{n}+\mathrm{2}}{\mathrm{4}}\right)+\mathrm{E}\left(\frac{{n}+\mathrm{4}}{\mathrm{4}}\right) \\ $$$$\mathrm{1st}\:\mathrm{case}\::\:{n}\:=\:\mathrm{4}{p} \\…
Question Number 144460 by byaw last updated on 25/Jun/21 $$ \\ $$$$\mathrm{If}\:\mathrm{C}=\mathrm{226}{k}\:\mathrm{find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{integal}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{that}\:\mathrm{will} \\ $$$$\mathrm{make}\:\mathrm{C}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square} \\ $$ Answered by Rasheed.Sindhi last updated on 26/Jun/21…
Question Number 13346 by tawa tawa last updated on 19/May/17 Commented by ajfour last updated on 19/May/17 $${your}\:{previous}\:{question}\:{Q}\mathrm{13345} \\ $$$${something}\:{wrong}\:{uploading}\:{it}. \\ $$$${cannot}\:{click}\:{to}\:{answer}.. \\ $$ Commented…
Question Number 144342 by ArielVyny last updated on 25/Jun/21 $${Montrer}\:{que}\:\forall{n}\in\mathbb{Z} \\ $$$${E}\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)+{E}\left(\frac{{n}+\mathrm{2}}{\mathrm{4}}\right)+{E}\left(\frac{{n}+\mathrm{4}}{\mathrm{4}}\right)={n} \\ $$ Answered by Olaf_Thorendsen last updated on 24/Jun/21 $$\mathrm{Selon}\:\mathrm{moi},\:\mathrm{cette}\:\mathrm{formule}\:\mathrm{est}\:\mathrm{fausse}. \\ $$$$\mathrm{According}\:\mathrm{to}\:\mathrm{me},\:\mathrm{this}\:\mathrm{formula}\:\mathrm{is}\:\mathrm{false}. \\…
Question Number 78800 by Tony Lin last updated on 20/Jan/20 $${Simplify}: \\ $$$$\underset{\mathrm{3}} {\:}\sqrt{\mathrm{3}+\frac{\mathrm{10}}{\mathrm{3}}\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}{i}}+\underset{\mathrm{3}} {\:}\sqrt{\mathrm{3}−\frac{\mathrm{10}}{\mathrm{3}}\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}{i}} \\ $$ Commented by mathmax by abdo last updated on…