Question Number 100178 by bobhans last updated on 25/Jun/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{positif}\: \\ $$$$\mathrm{integers}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{that}\:\mathrm{satisfy}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{xy}=\mathrm{37} \\ $$ Answered by 1549442205 last updated on 25/Jun/20 $$\mathrm{x}^{\mathrm{2}} −\mathrm{xy}+\mathrm{y}^{\mathrm{2}}…
Question Number 100042 by bobhans last updated on 24/Jun/20 Answered by Rasheed.Sindhi last updated on 25/Jun/20 $$\:\:\:\mathrm{16}{p}={r}^{\mathrm{3}} −\mathrm{1} \\ $$$$\:\:\:\:\mathrm{16}{p}=\left({r}−\mathrm{1}\right)\left({r}^{\mathrm{2}} +{r}+\mathrm{1}\right) \\ $$$$\:\:\:\:{r}−\mathrm{1}=\mathrm{1},\mathrm{2},\mathrm{4},\mathrm{8},\mathrm{16},{p} \\ $$$$\:{r}=\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{9},\mathrm{17},{p}+\mathrm{1}\:\left(\mathrm{possible}\:\mathrm{values}\right)\:\:\:…
Question Number 34385 by Rasheed.Sindhi last updated on 05/May/18 $$\mathrm{p},\mathrm{q}\in\mathbb{P} \\ $$$$\mathrm{m},\mathrm{n}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…\right\} \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{are}\:\mathrm{there},\mathrm{whose} \\ $$$$\mathrm{LCM}\:\mathrm{is}\:\mathrm{p}^{\mathrm{m}} \mathrm{q}^{\mathrm{n}\:} ,\mathrm{when}: \\ $$$$\left(\mathrm{i}\right)\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{different}. \\ $$$$\:\:\:\:\left(\mathrm{Generalization}\:\mathrm{of}\:\mathrm{Q}#\:\mathrm{34358}\right) \\…
Question Number 34369 by Rasheed.Sindhi last updated on 05/May/18 $$\mathrm{Determine}\:\mathrm{number}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{pairs},\mathrm{whose} \\ $$$$\mathrm{GCD}\:\mathrm{is}\:\mathrm{144}\:\mathrm{in}\:\mathrm{case}: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{when}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{and}\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{is}\:\mathrm{considerd} \\ $$$$\:\:\:\:\:\:\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{when}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{and}\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{is}\:\mathrm{considerd} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{different}. \\ $$ Answered by MJS…
Question Number 34358 by Rasheed.Sindhi last updated on 05/May/18 $$\mathrm{Determine}\:\mathrm{number}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{pairs}\:\mathrm{whose} \\ $$$$\mathrm{LCM}\:\mathrm{is}\:\mathrm{144}\:\mathrm{in}\:\mathrm{case}, \\ $$$$\left(\mathrm{i}\right)\mathrm{when}\:\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\mathrm{when}\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{different}. \\ $$ Commented by candre last updated on 05/May/18…
Question Number 99804 by bramlex last updated on 23/Jun/20 $${Determine}\:{x},{y}\:\in\:\mathbb{Z}\:{such}\:{that}\: \\ $$$$\mathrm{1}+\mathrm{2}^{{x}} \:+\mathrm{2}^{\mathrm{2}{x}+\mathrm{1}} \:=\:{y}^{\mathrm{2}} \: \\ $$ Commented by Rasheed.Sindhi last updated on 23/Jun/20 $${Q}#\mathrm{97746}…
Question Number 34186 by Joel578 last updated on 02/May/18 $$\mathrm{Let}\:{x}_{\mathrm{1}} \:=\:\mathrm{0},\:{x}_{\mathrm{2}} \:=\:\mathrm{1}\:\mathrm{and}\:{x}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}_{{n}−\mathrm{1}} \:+\:{x}_{{n}−\mathrm{2}} \right) \\ $$$$\mathrm{Show}\:\mathrm{that}\: \\ $$$${x}_{{n}} \:=\:\frac{\mathrm{2}^{{n}−\mathrm{1}} \:+\:\left(−\mathrm{1}\right)^{{n}} }{\mathrm{3}\:.\:\mathrm{2}^{{n}−\mathrm{2}} } \\ $$…
Question Number 99495 by bobhans last updated on 21/Jun/20 $$\mathrm{solve}\:\mathrm{for}\:\mathrm{x},\mathrm{y}\:\in\:\mathbb{N}\: \\ $$$$\mathrm{7}^{\mathrm{y}} +\mathrm{2}\:=\:\mathrm{3}^{\mathrm{x}} \: \\ $$ Commented by PRITHWISH SEN 2 last updated on 21/Jun/20…
Question Number 164940 by amin96 last updated on 23/Jan/22 $$−−−−−−−−− \\ $$$$\mathrm{1}!−\mathrm{2}!+\mathrm{3}!−\mathrm{4}!+\mathrm{5}!−\ldots−\mathrm{14}!+\mathrm{15}!=? \\ $$$$ \\ $$$$−−−−−−−−−−\boldsymbol{{by}}\:\boldsymbol{{M}}.\boldsymbol{{A}} \\ $$ Commented by MJS_new last updated on 24/Jan/22…
Question Number 99322 by bramlex last updated on 20/Jun/20 $${what}\:{is}\:{remainder}\:{of}\:\mathrm{2}^{\mathrm{7}^{\mathrm{2002}} } \:{divided} \\ $$$${by}\:\mathrm{352}\: \\ $$ Answered by john santu last updated on 20/Jun/20 Commented…