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Category: Algebra

The-number-of-pairs-of-natural-numbers-x-y-satisfy-x-2-y-gcd-x-y-2-2023-

Question Number 189531 by cortano12 last updated on 18/Mar/23 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{satisfy} \\ $$$$\:\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \mathrm{y}+\mathrm{gcd}\left(\mathrm{x},\mathrm{y}^{\mathrm{2}} \right)=\mathrm{2023} \\ $$ Answered by aleks041103 last updated on 19/Mar/23…

Prove-without-mathematical-induction-that-the-expression-1-2-2n-1-2-2n-is-even-for-every-natural-number-n-

Question Number 58409 by Tawa1 last updated on 22/Apr/19 $$\mathrm{Prove}\:\mathrm{without}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{expression}\:\:\:\left(\mathrm{1}\:+\:\sqrt{\mathrm{2}}\right)^{\mathrm{2n}} \:+\:\left(\mathrm{1}\:−\:\sqrt{\mathrm{2}}\right)^{\mathrm{2n}} \:\:\mathrm{is}\:\mathrm{even}\:\mathrm{for}\:\mathrm{every} \\ $$$$\mathrm{natural}\:\mathrm{number}\:\:\mathrm{n}. \\ $$ Commented by maxmathsup by imad last updated…

Show-that-the-sum-of-the-cube-of-three-consecutive-number-gives-a-multiple-of-9-

Question Number 58410 by Tawa1 last updated on 22/Apr/19 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{of}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{number}\:\mathrm{gives}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\:\mathrm{9}. \\ $$ Answered by mr W last updated on 22/Apr/19 $${S}=\left({n}−\mathrm{1}\right)^{\mathrm{3}} +{n}^{\mathrm{3}} +\left({n}+\mathrm{1}\right)^{\mathrm{3}}…

Question-189473

Question Number 189473 by Spillover last updated on 17/Mar/23 Answered by a.lgnaoui last updated on 17/Mar/23 $${x}=\mathrm{1}\:\:\:{f}\left(\mathrm{1}\right)=\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{3}} =\frac{\mathrm{27}}{\mathrm{64}} \\ $$$${x}=\mathrm{2}\:\:{f}\left(\mathrm{2}\right)=\left(\frac{\mathrm{3}}{\mathrm{8}}\right)^{\mathrm{2}} =\frac{\mathrm{9}}{\mathrm{64}} \\ $$$${x}=\mathrm{3}\:\:{f}\left(\mathrm{3}\right)=\left(\frac{\mathrm{3}}{\mathrm{12}}\right)^{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{4}}=\frac{\mathrm{16}}{\mathrm{64}} \\…

The-imaginary-part-of-1-2-1-2-i-10-is-

Question Number 58402 by rahul 19 last updated on 22/Apr/19 $${The}\:{imaginary}\:{part}\:{of}\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{i}\right)^{\mathrm{10}} {is}\:? \\ $$ Commented by maxmathsup by imad last updated on 23/Apr/19 $${let}\:{Z}\:=\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{i}\right)^{\mathrm{10}} \:\Rightarrow{Z}\:=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{10}}…