Question Number 127979 by bobhans last updated on 03/Jan/21 $$\:\:\left(\mathrm{1}+{i}\right)^{\mathrm{2020}} \:=? \\ $$ Answered by liberty last updated on 03/Jan/21 $$\:\left(\mathrm{1}+\mathrm{i}\right)^{\mathrm{2020}} \:=\:\left(\sqrt{\mathrm{2}}\:\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:}\:+\:\frac{\mathrm{i}}{\:\sqrt{\mathrm{2}}}\right)\right)^{\mathrm{2020}} \\ $$$$\:=\:\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2020}} \:\left(\mathrm{cos}\:\left(\frac{\pi}{\mathrm{4}}\right)+\mathrm{i}\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}\right)\right)^{\mathrm{2020}}…
Question Number 127940 by bramlexs22 last updated on 03/Jan/21 $$\:\begin{cases}{\mathrm{3x}=\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{4}\right)}\\{\mathrm{4x}=\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{5}\right)\:}\\{\mathrm{5x}=\mathrm{7}\:\left(\mathrm{mod}\:\mathrm{11}\right)}\end{cases} \\ $$ Answered by floor(10²Eta[1]) last updated on 03/Jan/21 $$\mathrm{3x}\equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{4}\right)\Rightarrow\mathrm{x}\equiv\mathrm{3}\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\mathrm{x}=\mathrm{4a}+\mathrm{3} \\ $$$$\mathrm{4}\left(\mathrm{4a}+\mathrm{3}\right)\equiv\mathrm{3}\left(\mathrm{mod}\:\mathrm{5}\right) \\…
Question Number 62244 by Rasheed.Sindhi last updated on 18/Jun/19 $$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y},\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\mathrm{gcd}\left(\mathrm{x}^{\mathrm{3}} ,\mathrm{y}^{\mathrm{2}} \right)=\mathrm{gcd}\left(\mathrm{x}^{\mathrm{2}} ,\mathrm{y}^{\mathrm{3}} \right) \\ $$ Commented by mr W last updated on…
Question Number 62242 by Rasheed.Sindhi last updated on 18/Jun/19 $$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{180}\:\wedge\:\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{45} \\ $$ Commented by maxmathsup by imad last updated on 19/Jun/19 $$\Delta\left({x},{y}\right)\:=\mathrm{45}\:{and}\:{M}\left({x},{y}\right)=\mathrm{180}\:\Rightarrow{x}\:=\mathrm{45}\:{u}\:\:{and}\:{y}\:=\mathrm{45}\:{v}\:{with}\:\Delta\left({u},{v}\right)=\mathrm{1} \\…
Question Number 62214 by Rasheed.Sindhi last updated on 17/Jun/19 $$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)}=\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)−\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right) \\ $$ Answered by MJS last updated on 17/Jun/19 $$\mathrm{lcm}\:\left({x},\:{y}\right)=\frac{\mathrm{gcd}^{\mathrm{2}} \:\left({x},\:{y}\right)}{\mathrm{gcd}\:\left({x},{y}\right)\:−\mathrm{1}} \\ $$$$\Rightarrow\:\mathrm{gcd}\:\left({x},{y}\right)=\mathrm{2}\wedge\mathrm{lcm}\:\left({x},\:{y}\right)=\mathrm{4}\:\Rightarrow\:…
Question Number 127610 by pticantor last updated on 31/Dec/20 $$\boldsymbol{{Z}}=\mathrm{1}+\left(\mathrm{1}+{i}\right)\boldsymbol{{cos}\theta} \\ $$$$\boldsymbol{{arg}}\left(\boldsymbol{{z}}\right)=? \\ $$ Answered by MJS_new last updated on 31/Dec/20 $$\mathrm{arg}\:\left(\mathrm{1}+\mathrm{cos}\:\theta\:+\mathrm{i}\:\mathrm{cos}\:\theta\right)\:= \\ $$$$=\frac{\pi}{\mathrm{2}}\mathrm{sign}\:\left(\mathrm{cos}\:\theta\right)\:−\mathrm{arctan}\:\frac{\mathrm{1}+\mathrm{cos}\:\theta}{\mathrm{cos}\:\theta} \\…
Question Number 192909 by uchihayahia last updated on 30/May/23 $$\: \\ $$$$\:{let}\:{P}\left({x}\right)\:{is}\:{polinomial}\:{with}\:{integer} \\ $$$$\:{coefficient}\:{s}.{t}\:{P}\left(\mathrm{6}\right){P}\left(\mathrm{38}\right){P}\left(\mathrm{57}\right)+\mathrm{19}\:{is} \\ $$$$\:{divided}\:{by}\:\mathrm{114}.\:{P}\left(-\mathrm{13}\right)=\mathrm{479}\:{and}\:{P}\geqslant\mathrm{0} \\ $$$$\:{what}\:{is}\:{minimum}\:{value}\:{of}\:{P}\left(\mathrm{0}\right)? \\ $$$$ \\ $$ Terms of Service…
Question Number 127111 by benjo_mathlover last updated on 26/Dec/20 $$\:{find}\:{the}\:{value}\:{of}\:{x}\:{such}\:{that}\: \\ $$$$\:\begin{cases}{{x}=\mathrm{2}\:\left({mod}\:\mathrm{5}\right)}\\{{x}=\mathrm{3}\:\left({mod}\:\mathrm{8}\right)\:}\\{{x}=\mathrm{2}\:\left({mod}\:\mathrm{3}\right)}\end{cases} \\ $$ Answered by liberty last updated on 04/Jan/21 $${given}\:\begin{cases}{{x}=\mathrm{2}\:\left({mod}\:\mathrm{5}\right)…\left({i}\right)}\\{{x}=\mathrm{3}\:\left({mod}\:\mathrm{8}\right)…\left({ii}\right)}\\{{x}=\mathrm{2}\:\left({mod}\:\mathrm{3}\right)…\left({iii}\right)}\end{cases} \\ $$$${for}\left({i}\right)\:\Rightarrow\:\mathrm{24}{a}\:\equiv\:\mathrm{2}\:\left({mod}\:\mathrm{5}\right)\: \\…
Question Number 126682 by bramlexs22 last updated on 23/Dec/20 $$\:{Find}\:{the}\:{remainder}\: \\ $$$$\:\mathrm{7}+\mathrm{77}+\mathrm{777}+\mathrm{7777}+…+\underset{\mathrm{2020}\:{times}} {\underbrace{\mathrm{777}…\mathrm{7}}} \\ $$$${when}\:{divided}\:{by}\:\mathrm{9}. \\ $$ Answered by talminator2856791 last updated on 23/Dec/20 $$\:…
Question Number 126677 by AST last updated on 26/Sep/22 Commented by talminator2856791 last updated on 23/Dec/20 $$\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{question}? \\ $$$$\: \\ $$ Commented by talminator2856791 last…