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Question Number 144989    Answers: 1   Comments: 0

lcm(2a;3a)=lcm(45;100)⇒a=?

$${lcm}\left(\mathrm{2}{a};\mathrm{3}{a}\right)={lcm}\left(\mathrm{45};\mathrm{100}\right)\Rightarrow{a}=? \\ $$

Question Number 144985    Answers: 1   Comments: 0

∫ (((2+(√x)))/((x+1+(√x))^2 )) dx =?

$$\:\int\:\frac{\left(\mathrm{2}+\sqrt{\mathrm{x}}\right)}{\left(\mathrm{x}+\mathrm{1}+\sqrt{\mathrm{x}}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 144981    Answers: 1   Comments: 0

Σ_(k=1) ^(35) ((√k)/(k + (√(k^2 + k)))) = ?

$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{35}} {\sum}}\:\frac{\sqrt{{k}}}{{k}\:+\:\sqrt{{k}^{\mathrm{2}} \:+\:{k}}}\:=\:? \\ $$

Question Number 144980    Answers: 1   Comments: 0

exercise Let a and b be natural integers such that 0<a<b. 1. Show that if a divides b, then for any naturel number n, n^a −1 divides n^b −1. 2. For any non−zero naturel number n, prove that the remainder of the euclidean division of n^b −1 by n^a −1 is n^r −1 where r is the remainder of the euclidean division of b by a. 3. For any non−zero naturel number n, show that gcd(n^b −1, n^a −1) = n^d −1 where d = gcd(b,c). by professor henderson^(−) .

$$\underline{\boldsymbol{\mathrm{exercise}}} \\ $$$$\boldsymbol{\mathrm{Let}}\:\boldsymbol{{a}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{b}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{natural}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{{a}}\:\boldsymbol{\mathrm{divides}}\:\boldsymbol{{b}},\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{naturel}}\: \\ $$$$\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{{n}}^{\boldsymbol{{a}}} −\mathrm{1}\:\boldsymbol{\mathrm{divides}}\:\boldsymbol{{n}}^{\boldsymbol{{b}}} −\mathrm{1}. \\ $$$$ \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{For}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{non}}−\boldsymbol{\mathrm{zero}}\:\boldsymbol{\mathrm{naturel}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{\mathrm{prove}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{remainder}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{euclidean}}\:\boldsymbol{\mathrm{division}}\:\boldsymbol{\mathrm{of}}\: \\ $$$$\boldsymbol{{n}}^{\boldsymbol{{b}}} −\mathrm{1}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{{n}}^{\boldsymbol{{a}}} −\mathrm{1}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{{n}}^{\boldsymbol{{r}}} −\mathrm{1}\:\boldsymbol{\mathrm{where}}\:\boldsymbol{{r}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{remainder}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{euclidean}}\:\boldsymbol{\mathrm{division}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{b}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{{a}}. \\ $$$$ \\ $$$$\mathrm{3}.\:\boldsymbol{\mathrm{For}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{non}}−\boldsymbol{\mathrm{zero}}\:\boldsymbol{\mathrm{naturel}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{{n}},\:\boldsymbol{\mathrm{show}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{{gcd}}\left(\boldsymbol{{n}}^{\boldsymbol{{b}}} −\mathrm{1},\:\boldsymbol{{n}}^{\boldsymbol{{a}}} −\mathrm{1}\right)\:=\:\boldsymbol{{n}}^{\boldsymbol{{d}}} −\mathrm{1}\:\boldsymbol{\mathrm{where}}\:\boldsymbol{{d}}\:=\:\boldsymbol{{gcd}}\left(\boldsymbol{{b}},\boldsymbol{{c}}\right). \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overline {\boldsymbol{{by}}\:\boldsymbol{{professor}}\:\boldsymbol{{henderson}}}. \\ $$

Question Number 144975    Answers: 3   Comments: 0

Question Number 144959    Answers: 2   Comments: 0

Question Number 144961    Answers: 1   Comments: 0

x∈Z^+ 15^(48a+1) ≡ x (mod 17) find x=?

$${x}\in\mathbb{Z}^{+} \\ $$$$\mathrm{15}^{\mathrm{48}\boldsymbol{{a}}+\mathrm{1}} \:\equiv\:{x}\:\left({mod}\:\mathrm{17}\right)\:\:{find}\:\:{x}=?\: \\ $$

Question Number 144951    Answers: 0   Comments: 1

Let a,b > 0 and a+b+1 = 3ab. Prove that (1) (a^2 /(a+1))+(b^2 /(b+1)) ≥ (a/(a^2 +1))+(b/(b^2 +1)) (2) (a^2 /(b+1))+(b^2 /(a+1)) ≥ (a/(b^2 +1))+(b/(a^2 +1))

$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} }{{a}+\mathrm{1}}+\frac{{b}^{\mathrm{2}} }{{b}+\mathrm{1}}\:\geqslant\:\frac{{a}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{b}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} }{{b}+\mathrm{1}}+\frac{{b}^{\mathrm{2}} }{{a}+\mathrm{1}}\:\geqslant\:\frac{{a}}{{b}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{a}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 144947    Answers: 1   Comments: 0

Π_(k=1) ^(12) 2∙sin(((πk)/(24))) = ?

$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{12}} {\prod}}\mathrm{2}\centerdot{sin}\left(\frac{\pi{k}}{\mathrm{24}}\right)\:=\:? \\ $$

Question Number 144946    Answers: 0   Comments: 0

𝛗 := ∫_( 0) ^( (π/2)) ((( x)/(cot ( x ))) )^( 3) dx=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}\::=\:\int_{\:\mathrm{0}} ^{\:\:\frac{\pi}{\mathrm{2}}} \left(\frac{\:\mathrm{x}}{\mathrm{cot}\:\left(\:\mathrm{x}\:\right)}\:\right)^{\:\mathrm{3}} \mathrm{dx}=? \\ $$$$ \\ $$

Question Number 144939    Answers: 0   Comments: 0

resoudre I=∫_0 ^(π/2) tan(nx)tan^n (x)dx

$$ \\ $$$$\boldsymbol{\mathrm{resoudre}} \\ $$$$\boldsymbol{\mathrm{I}}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \boldsymbol{\mathrm{tan}}\left(\boldsymbol{\mathrm{nx}}\right)\boldsymbol{\mathrm{tan}}^{\boldsymbol{\mathrm{n}}} \left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}} \\ $$

Question Number 144936    Answers: 1   Comments: 0

Question Number 144930    Answers: 2   Comments: 0

solve (d^2 x/dt^2 )=cosx

$${solve} \\ $$$$\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }={cosx} \\ $$

Question Number 144929    Answers: 1   Comments: 0

find the value lim_(n→∞) (1+((1+(1/2)+(1/3)+(1/4)+...+(1/n))/n^2 ))^n

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\: \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+...+\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{n}^{\mathrm{2}} }\right)^{\mathrm{n}} \:\: \\ $$

Question Number 144928    Answers: 1   Comments: 0

Σ_(n≥0) (((−1)^n )/((n+z)n!))

$$\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+{z}\right){n}!} \\ $$

Question Number 144926    Answers: 1   Comments: 0

∫_0 ^∞ x^n (e^(ix) )^z dx=??? (z∈C)

$$\int_{\mathrm{0}} ^{\infty} {x}^{{n}} \left({e}^{{ix}} \right)^{{z}} {dx}=???\:\:\:\left({z}\in\mathbb{C}\right) \\ $$

Question Number 144925    Answers: 1   Comments: 0

Σ_(n=1) ^∞ (((2n)!!)/(2^n ∙(n+1)∙(2n+1)!!))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2n}\right)!!}{\mathrm{2}^{\mathrm{n}} \centerdot\left(\mathrm{n}+\mathrm{1}\right)\centerdot\left(\mathrm{2n}+\mathrm{1}\right)!!}=? \\ $$

Question Number 144924    Answers: 1   Comments: 0

S(x)=Σ_(n=1) ^∞ (((2n)!!)/((2n+1)!!))x^(2n) =?........(∣x∣<1)

$$\mathrm{S}\left(\mathrm{x}\right)=\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2n}\right)!!}{\left(\mathrm{2n}+\mathrm{1}\right)!!}\mathrm{x}^{\mathrm{2n}} =?........\left(\mid\mathrm{x}\mid<\mathrm{1}\right) \\ $$

Question Number 144922    Answers: 1   Comments: 0

if z^2 - 16(√z) = 12 find z - 2(√z) = ?

$${if}\:\:{z}^{\mathrm{2}} \:-\:\mathrm{16}\sqrt{{z}}\:=\:\mathrm{12} \\ $$$${find}\:\:{z}\:-\:\mathrm{2}\sqrt{{z}}\:=\:? \\ $$

Question Number 144917    Answers: 1   Comments: 0

if x;y>0 then: 10 ∙ (√((x^2 +y^2 )/2)) + ((8xy)/(x+y)) ≥ 7x+7y

$${if}\:\:{x};{y}>\mathrm{0}\:\:{then}: \\ $$$$\mathrm{10}\:\centerdot\:\sqrt{\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{\mathrm{2}}}\:+\:\frac{\mathrm{8}{xy}}{{x}+{y}}\:\geqslant\:\mathrm{7}{x}+\mathrm{7}{y} \\ $$

Question Number 144914    Answers: 2   Comments: 0

If ((1+tan 4(√θ))/(1−tan 4(√θ))) = tan θ , then find possible value of tan (θ+11(√θ) ).

$$\:\mathrm{If}\:\frac{\mathrm{1}+\mathrm{tan}\:\mathrm{4}\sqrt{\theta}}{\mathrm{1}−\mathrm{tan}\:\mathrm{4}\sqrt{\theta}}\:=\:\mathrm{tan}\:\theta\:,\:\mathrm{then}\:\mathrm{find}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{tan}\:\left(\theta+\mathrm{11}\sqrt{\theta}\:\right). \\ $$

Question Number 144910    Answers: 0   Comments: 0

Γ(((n+1)/(1−i)))=????

$$\Gamma\left(\frac{{n}+\mathrm{1}}{\mathrm{1}−{i}}\right)=???? \\ $$

Question Number 144909    Answers: 1   Comments: 0

Γ(a+ib) doesn′t exist ? give her value

$$\Gamma\left({a}+{ib}\right)\:{doesn}'{t}\:{exist}\:?\:{give}\:{her}\:{value} \\ $$

Question Number 144903    Answers: 0   Comments: 0

Question Number 144900    Answers: 1   Comments: 0

etude complete de la courbe d′equation polaire r=(1/(sin(2θ))) (symetrie et trace)

$${etude}\:{complete}\:{de}\:{la}\:{courbe}\:{d}'{equation} \\ $$$${polaire}\:{r}=\frac{\mathrm{1}}{{sin}\left(\mathrm{2}\theta\right)}\:\:\:\:\left({symetrie}\:{et}\:{trace}\right) \\ $$$$ \\ $$

Question Number 144899    Answers: 0   Comments: 0

Ω := ∫ ((√(1−sin(x)))/(cos (x))) e^(−(1/2) x) = ?

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\Omega\::=\:\int\:\frac{\sqrt{\mathrm{1}−{sin}\left({x}\right)}}{{cos}\:\left({x}\right)}\:{e}\:^{−\frac{\mathrm{1}}{\mathrm{2}}\:{x}} =\:? \\ $$$$ \\ $$

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