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Question Number 216875 Answers: 1 Comments: 0

Let p be a prime number greater than 3. Prove that p^2 − 1 is always divisible by 24.

$$\mathrm{Let}\:\:\mathrm{p}\:\:\mathrm{be}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{p}^{\mathrm{2}} −\:\mathrm{1}\:\: \\ $$$$\mathrm{is}\:\:\mathrm{always}\:\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{24}. \\ $$

Question Number 216861 Answers: 0 Comments: 0

Question Number 216859 Answers: 1 Comments: 3

Question Number 216855 Answers: 2 Comments: 0

Question Number 216842 Answers: 1 Comments: 0

Find all pairs of positive integers x, y that satisfy the system xy + x + y=71 x^2 y + xy^2 =880

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{x},\:\mathrm{y}\:\:\mathrm{that}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\:\mathrm{system}\:\: \\ $$$$\mathrm{xy}\:+\:\mathrm{x}\:+\:\mathrm{y}=\mathrm{71}\: \\ $$$$\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}^{\mathrm{2}} =\mathrm{880} \\ $$

Question Number 216841 Answers: 1 Comments: 0

Prove that: δ(n) = Σ_(d/n) 𝛟(d) 𝛕((n/d)) 𝛅(n) = Σ_(d/n) d , 𝛕(n) = Σ_(d/n) l and ϕ-Eyler.f

$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\:\:\delta\left(\mathrm{n}\right)\:=\:\underset{\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{n}}}} {\sum}\:\boldsymbol{\varphi}\left(\mathrm{d}\right)\:\boldsymbol{\tau}\left(\frac{\mathrm{n}}{\mathrm{d}}\right) \\ $$$$\boldsymbol{\delta}\left(\mathrm{n}\right)\:=\:\underset{\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{n}}}} {\sum}\:\mathrm{d}\:\:\:,\:\:\:\boldsymbol{\tau}\left(\mathrm{n}\right)\:=\:\underset{\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{n}}}} {\sum}\:{l}\:\:\:\mathrm{and}\:\:\:\varphi-\mathrm{Eyler}.\mathrm{f} \\ $$

Question Number 216836 Answers: 3 Comments: 0

Find: (((1 + tan1°)(1 + tan2°)...(1 + tan44°))/((1−tan46°)(1−tan47°)...(1−tan89°))) = ?

$$\mathrm{Find}: \\ $$$$\frac{\left(\mathrm{1}\:+\:\mathrm{tan1}°\right)\left(\mathrm{1}\:+\:\mathrm{tan2}°\right)...\left(\mathrm{1}\:+\:\mathrm{tan44}°\right)}{\left(\mathrm{1}−\mathrm{tan46}°\right)\left(\mathrm{1}−\mathrm{tan47}°\right)...\left(\mathrm{1}−\mathrm{tan89}°\right)}\:=\:? \\ $$

Question Number 216830 Answers: 1 Comments: 1

Prove:∀x∈R,∣cos x∣+∣cos 2x∣+…+∣cos nx∣≥((n−1)/2)(n∈Z_(>0) )

$$\mathrm{Prove}:\forall{x}\in\mathbb{R},\mid\mathrm{cos}\:{x}\mid+\mid\mathrm{cos}\:\mathrm{2}{x}\mid+\ldots+\mid\mathrm{cos}\:{nx}\mid\geq\frac{{n}−\mathrm{1}}{\mathrm{2}}\left({n}\in\mathbb{Z}_{>\mathrm{0}} \right)\:\: \\ $$

Question Number 216827 Answers: 1 Comments: 1

Question Number 216821 Answers: 0 Comments: 0

Question Number 216820 Answers: 2 Comments: 0

f(x) = ax^4 + bx^3 + cx^2 + dx + e f(1) = 2 f(2) = 3 f(3) = 4 f(4) = 5 f(0) = 25 Then f(5) = ? Help me, please

$$ \\ $$$$\:\:\:{f}\left({x}\right)\:=\:{ax}^{\mathrm{4}} \:+\:{bx}^{\mathrm{3}} \:+\:{cx}^{\mathrm{2}} \:+\:{dx}\:+\:{e} \\ $$$$\:\:\:{f}\left(\mathrm{1}\right)\:=\:\mathrm{2} \\ $$$$\:\:\:{f}\left(\mathrm{2}\right)\:=\:\mathrm{3} \\ $$$$\:\:\:{f}\left(\mathrm{3}\right)\:=\:\mathrm{4} \\ $$$$\:\:\:{f}\left(\mathrm{4}\right)\:=\:\mathrm{5}\: \\ $$$$\:\:\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{25} \\ $$$$\:\:\:\mathcal{T}{hen}\:\:{f}\left(\mathrm{5}\right)\:=\:? \\ $$$$\:\:\:\mathcal{H}{elp}\:{me},\:\:{please} \\ $$$$ \\ $$

Question Number 216819 Answers: 1 Comments: 0

Prove:∫_(0 ) ^1 ((K(x))/( (√(3−x))))dx=(1/(96π(√3)))×Γ((1/(24)))Γ((3/(24)))Γ((7/(24)))Γ(((11)/(24)))

$$\mathrm{Prove}:\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{K}}\left({x}\right)}{\:\sqrt{\mathrm{3}−{x}}}{dx}=\frac{\mathrm{1}}{\mathrm{96}\pi\sqrt{\mathrm{3}}}×\Gamma\left(\frac{\mathrm{1}}{\mathrm{24}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{24}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{24}}\right)\Gamma\left(\frac{\mathrm{11}}{\mathrm{24}}\right) \\ $$

Question Number 216810 Answers: 0 Comments: 1

40 random numbers picked from 0 to 100. what is the probability that at least half of them has the range of 10.

$$ \\ $$40 random numbers picked from 0 to 100. what is the probability that at least half of them has the range of 10.

Question Number 216800 Answers: 1 Comments: 0

Question Number 216799 Answers: 1 Comments: 1

Question Number 216807 Answers: 1 Comments: 0

Uh guys is the speed formula (d/t) or lim_(Δt→0) ((Δd)/(Δt))

$$\mathrm{Uh}\:\mathrm{guys}\:\mathrm{is}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{formula} \\ $$$$\frac{{d}}{{t}} \\ $$$$\mathrm{or} \\ $$$$\mathrm{li}\underset{\Delta{t}\rightarrow\mathrm{0}} {\mathrm{m}}\frac{\Delta{d}}{\Delta{t}} \\ $$

Question Number 216792 Answers: 0 Comments: 0

Question Number 216788 Answers: 1 Comments: 1

Question Number 216787 Answers: 1 Comments: 0

form the differential equationfrom the following 1) y=Ae^(3x) +Be^(5x) 2) y^2 =(x−1) 3) c(y+c)^2 +x^3 =0

$${form}\:{the}\:{differential}\:{equationfrom}\:{the}\:{following} \\ $$$$\left.\mathrm{1}\right)\:{y}={Ae}^{\mathrm{3}{x}} +{Be}^{\mathrm{5}{x}} \\ $$$$\left.\mathrm{2}\right)\:{y}^{\mathrm{2}} =\left({x}−\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{c}\left({y}+{c}\right)^{\mathrm{2}} +{x}^{\mathrm{3}} =\mathrm{0} \\ $$

Question Number 216786 Answers: 1 Comments: 0

Question Number 216785 Answers: 0 Comments: 0

given the recursive {a_n } define by setting a_(1 ) ∈ (0,1) , a_(n+1) = a_n (1−a_n ) , n≥1 prove that (1) lim_(n→∞) na_n = 1 (2) b_n = n(1−na_n ) is a incresing sequence and diverge to ∞ (3) lim_(n→∞) ((n(1−na_n ))/(ln(n))) = 1

$$\:\:\:\mathrm{given}\:\mathrm{the}\:\mathrm{recursive}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}\:\mathrm{define}\:\mathrm{by}\:\mathrm{setting} \\ $$$$\:\:\mathrm{a}_{\mathrm{1}\:} \:\in\:\left(\mathrm{0},\mathrm{1}\right)\:\:\:,\:\:\:\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:=\:\mathrm{a}_{\mathrm{n}} \left(\mathrm{1}−\mathrm{a}_{\mathrm{n}} \right)\:\:\:,\:\mathrm{n}\geqslant\mathrm{1} \\ $$$$\:\:\mathrm{prove}\:\mathrm{that}\:\:\left(\mathrm{1}\right)\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{na}_{\mathrm{n}} =\:\mathrm{1} \\ $$$$\:\:\left(\mathrm{2}\right)\:\:\mathrm{b}_{\mathrm{n}} \:=\:\mathrm{n}\left(\mathrm{1}−\mathrm{na}_{\mathrm{n}} \right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{incresing}\:\mathrm{sequence} \\ $$$$\:\:\:\mathrm{and}\:\mathrm{diverge}\:\mathrm{to}\:\infty \\ $$$$\:\:\:\left(\mathrm{3}\right)\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{n}\left(\mathrm{1}−\mathrm{na}_{\mathrm{n}} \right)}{\mathrm{ln}\left(\mathrm{n}\right)}\:=\:\mathrm{1} \\ $$

Question Number 216783 Answers: 3 Comments: 0

Find all positive integers n such that n^2 +7n+6 is perfect square.

$${Find}\:{all}\:{positive}\:{integers}\:{n}\:{such}\:{that} \\ $$$${n}^{\mathrm{2}} +\mathrm{7}{n}+\mathrm{6}\:{is}\:{perfect}\:{square}. \\ $$

Question Number 216776 Answers: 1 Comments: 0

Question Number 216774 Answers: 1 Comments: 0

find ∫ ((tan^2 (x) )/(1+sec^4 (x))) .dx

$$\:\:\boldsymbol{{find}}\:\int\:\frac{\boldsymbol{{tan}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)\:}{\mathrm{1}+\boldsymbol{{sec}}^{\mathrm{4}} \left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}}\: \\ $$

Question Number 216772 Answers: 2 Comments: 0

find ∫((tan^2 (x) )/(1−sec^4 (x))) .dx

$$\:\:\boldsymbol{{find}}\:\int\frac{\boldsymbol{{tan}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)\:}{\mathrm{1}−\boldsymbol{{sec}}^{\mathrm{4}} \left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}}\:\: \\ $$

Question Number 216769 Answers: 1 Comments: 0

Solve for integer k,m and n: k^2 m−n^2 =8

$${Solve}\:{for}\:{integer}\:{k},{m}\:{and}\:{n}: \\ $$$${k}^{\mathrm{2}} {m}−{n}^{\mathrm{2}} =\mathrm{8} \\ $$

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