Question Number 170046 by Rasheed.Sindhi last updated on 14/May/22 $$\mathrm{PROVE}\:\mathrm{THAT} \\ $$$$\:\:\:\:\mathrm{6}\:\mid\:\left(\mathrm{8}^{{x}} −\mathrm{2}^{{x}} \right) \\ $$ Answered by JDamian last updated on 14/May/22 $$\left(\mathrm{8}^{{x}} −\mathrm{2}^{{x}}…
Question Number 170003 by mathlove last updated on 13/May/22 $${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} }=\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$ Commented by mathlove last updated on…
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Question Number 104139 by bobhans last updated on 19/Jul/20 $${what}\:{is}\:{remainder}\:{when}\:\mathrm{61}^{\mathrm{61}} \:{divided} \\ $$$${by}\:\mathrm{1001} \\ $$ Commented by aurpeyz last updated on 19/Jul/20 $$\mathrm{how}? \\ $$…
Question Number 104100 by bramlex last updated on 19/Jul/20 $$\left(\mathrm{1}\right)\mathbb{F}{ind}\:{all}\:{natural}\:{pairs}\:{of} \\ $$$${integers}\:\left({x},{y}\right)\:{such}\:{that}\:{x}^{\mathrm{3}} −{y}^{\mathrm{3}} ={xy}+\mathrm{61}. \\ $$$$\left(\mathrm{2}\right)\mathbb{F}{ind}\:{gcd}\left({x}^{\mathrm{4}} −{x}^{\mathrm{3}} ,\:{x}^{\mathrm{3}} −{x}\right)\: \\ $$ Answered by bemath last…
Question Number 104093 by bramlex last updated on 19/Jul/20 $$\left(\mathrm{1}\right){Evaluate}\:{the}\:{sum}\:\lfloor\frac{\mathrm{2}^{\mathrm{0}} }{\mathrm{3}}\rfloor+\lfloor\frac{\mathrm{2}^{\mathrm{1}} }{\mathrm{3}}\rfloor+\lfloor\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}}\rfloor+…+\lfloor\frac{\mathrm{2}^{\mathrm{1000}} }{\mathrm{3}}\rfloor \\ $$$$\left(\mathrm{2}\right)\:{find}\:\mathrm{2}^{\mathrm{98}} \:\left({mod}\:\mathrm{33}\right)\: \\ $$ Answered by JDamian last updated on…
Question Number 104038 by bramlex last updated on 19/Jul/20 $${what}\:{is}\:{the}\:{largest}\:{positive} \\ $$$${integer}\:{n}\:{such}\:{that}\:{n}^{\mathrm{3}} +\mathrm{100}\:{is} \\ $$$${divisible}\:{by}\:{n}+\mathrm{10}\:?\: \\ $$ Answered by john santu last updated on 19/Jul/20…
Question Number 169541 by amin96 last updated on 02/May/22 Answered by aleks041103 last updated on 02/May/22 $${That}\:{is}\:{a}\:{repost}\:{to}\:{Q}.\mathrm{161817}\:. \\ $$$${I}'{ve}\:{answered}\:{it}\:{there}. \\ $$ Terms of Service Privacy…
Question Number 38395 by NECx last updated on 25/Jun/18 $${Find}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${z}^{\mathrm{2}} +\mathrm{2}\left(\mathrm{1}+{i}\right){z}\:+\mathrm{2}=\mathrm{0} \\ $$$${leaving}\:{your}\:{answer}\:{in}\:{a}+{ib} \\ $$ Commented by maxmathsup by imad last updated on…
Question Number 103908 by bobhans last updated on 18/Jul/20 $${Find}\:{the}\:{least}\:{positive}\:{integer}\:{n}\:{for} \\ $$$${which}\:{there}\:{exists}\:{a}\:{set}\:\left\{\:{s}_{\mathrm{1}} ,{s}_{\mathrm{2}} ,{s}_{\mathrm{3}} ,…,{s}_{{n}} \:\right\} \\ $$$${consisting}\:{of}\:{n}\:{distinct}\:{positive}\:{integers} \\ $$$${such}\:{that}\:\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{\mathrm{1}} }\right)\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{\mathrm{2}} }\right)\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{\mathrm{3}} }\right)…\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{{n}} }\right) \\…