Question Number 2820 by Yozzis last updated on 27/Nov/15 $${We}\:{have}\:{the}\:{idea}\:{of}\:{Phythagorian}\:{triples} \\ $$$${as}\:{solutions}\:\left({x},{y},{z}\right)\:{to}\:{the}\:{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={z}^{\mathrm{2}} \\ $$$${where}\:{x},{y},{z}\in\mathbb{Z}^{+} .\: \\ $$$${How}\:{frequently}\:{do}\:{solutions}\:\left({x},{y},{z},{t}\right)\:\:{to}\:{the}\:{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}}…
Question Number 2815 by prakash jain last updated on 28/Nov/15 $$\eta\left({s}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{{s}} }\:\mathrm{Dirichlet}\:\mathrm{eta}\:\mathrm{function} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\eta\left({s}\right)=\left(\mathrm{1}−\mathrm{2}^{\mathrm{1}−{s}} \right)\zeta\left({s}\right) \\ $$ Commented by prakash…
Question Number 2699 by abcd last updated on 25/Nov/15 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\mathrm{3}^{\mathrm{215}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{43}. \\ $$ Answered by RasheedAhmad last updated on 25/Nov/15 $$ \\ $$$${Since}\:\left(\mathrm{3},\mathrm{43}\right)=\mathrm{1}\:{and}\:\mathrm{43}\:{is}\:{prime}…
Question Number 133761 by liberty last updated on 24/Feb/21 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{linear}\:\mathrm{congruence}\: \\ $$$$\mathrm{19}{x}\:\equiv\:\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{141}\:\right) \\ $$ Answered by bobhans last updated on 24/Feb/21 $${Using}\:{Euclidean}\:{algorithm}\: \\ $$$$\:\mathrm{141}\:=\:\mathrm{7}×\mathrm{19}+\mathrm{8} \\…
Question Number 133757 by liberty last updated on 24/Feb/21 Answered by bobhans last updated on 24/Feb/21 $${Let}\:{x}\:{be}\:{the}\:{least}\:{number}\:{of}\:{marbles} \\ $$$${in}\:{the}\:{box}\:,\:{such}\:{that}\:\begin{cases}{{x}\equiv\mathrm{5}\:\left({mod}\:\mathrm{7}\right)}\\{{x}\equiv\mathrm{6}\:\left({mod}\:\mathrm{11}\right)}\\{{x}\equiv\:\mathrm{8}\:\left({mod}\:\mathrm{13}\right)}\end{cases} \\ $$$${We}\:{can}\:{use}\:{Chinese}\:{remainder}\:{theorem} \\ $$$${a}_{\mathrm{1}} =\mathrm{5}\:;\:{a}_{\mathrm{2}} =\mathrm{6}\:;\:{a}_{\mathrm{3}}…
Question Number 2603 by Yozzis last updated on 23/Nov/15 $${You}\:{have}\:{a}\:\mathrm{3}\:{litre}\:{jug}\:{and}\:{a}\:\mathrm{5}\:{litre}\:{jug}.\:{Make}\:\mathrm{4}\:{litres}. \\ $$ Answered by RasheedAhmad last updated on 23/Nov/15 $$\mathrm{2}×\left(\mathrm{5}\:{litre}\:{jug}\right)−\mathrm{2}×\left(\mathrm{3}\:{litre}\:{jug}\right) \\ $$$$=\mathrm{4}\:{litres} \\ $$ Commented…
Question Number 2575 by prakash jain last updated on 22/Nov/15 $${a}_{\mathrm{1}} =\mathrm{0} \\ $$$${a}_{{n}} =\mathrm{27}×{a}_{{n}−\mathrm{1}} +\left({n}−\mathrm{1}\right) \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{m}} {\sum}}{a}_{{k}} =? \\ $$ Commented by…
Question Number 133611 by benjo_mathlover last updated on 23/Feb/21 $$\mathrm{Given}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation}\: \\ $$$$\begin{cases}{\mathrm{2x}−\mathrm{3y}\:=\:\mathrm{13}}\\{\mathrm{3x}+\mathrm{2y}\:=\:\mathrm{b}}\end{cases}\:,\:\mathrm{where}\:\mathrm{l}\:\leqslant\:\mathrm{b}\leqslant\:\mathrm{100}\:\mathrm{and} \\ $$$$\mathrm{b}\:\mathrm{is}\:\mathrm{integer}.\:\mathrm{Suppose}\:\mathrm{n}^{\mathrm{2}} \:=\:\mathrm{x}+\mathrm{y}\:\mathrm{where} \\ $$$$\mathrm{x},\mathrm{y}\:\mathrm{is}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{given}\:\mathrm{system}\: \\ $$$$\mathrm{of}\:\mathrm{equation}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n} \\ $$$$\mathrm{for}\:\mathrm{n}\:\mathrm{is}\:\mathrm{integer}\: \\ $$ Answered by…
Question Number 2432 by prakash jain last updated on 19/Nov/15 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{3333}^{\mathrm{4444}} , \\ $$$$\mathrm{Say}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{3333}^{\mathrm{4444}} \:\mathrm{is}\:\mathrm{A}, \\ $$$$\mathrm{If}\:\mathrm{A}>\mathrm{10}\:\mathrm{then}\:\mathrm{sum}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{A}. \\ $$$$\mathrm{This}\:\mathrm{process}\:\mathrm{is}\:\mathrm{repeated}\:\mathrm{until}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{digits}\:\mathrm{sum}\:{x}\:\mathrm{in}\:\mathrm{obtained}. \\ $$$${x}=? \\ $$…
Question Number 2387 by prakash jain last updated on 18/Nov/15 $$\mathrm{How}\:\mathrm{many}\:\mathrm{0}{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{1000}!? \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{digits}\:\mathrm{from}\:\mathrm{the}\:\mathrm{right}? \\ $$ Commented by prakash jain last updated on 18/Nov/15 $$\mathrm{Number}\:\mathrm{of}\:\mathrm{zero}\:\mathrm{can}\:\mathrm{be}\:\mathrm{computed}\:\mathrm{using} \\…