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Number TheoryQuestion and Answers: Page 14

Question Number 101043    Answers: 1   Comments: 0

((1+(1/2^(11) ) +(1/3^(11) ) +(1/4^(11) ) + ...)/(1−(1/2^(11) )+(1/3^(11) )−(1/4^(11) )+...)) =?

$$\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{11}} }\:+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{11}} }\:+\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{11}} }\:+\:...}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{11}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{11}} }−\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{11}} }+...}\:=? \\ $$

Question Number 100888    Answers: 0   Comments: 0

Find all pairs (k,n) of positive integer for which 7^k −3^n divides k^4 +n^2 .

$${Find}\:{all}\:{pairs}\:\left({k},{n}\right)\:{of}\:{positive}\: \\ $$$${integer}\:{for}\:{which}\:\mathrm{7}^{{k}} −\mathrm{3}^{{n}} \:{divides} \\ $$$${k}^{\mathrm{4}} +{n}^{\mathrm{2}} \:. \\ $$

Question Number 100733    Answers: 0   Comments: 6

A positive integer such as 4334 is a palindrome if it reads the same forwards or backwards. What is the only prime palindrome with an even number of digits?

$$\mathrm{A}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{as}\:\mathrm{4334}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{palindrome}\:\mathrm{if}\:\mathrm{it}\:\mathrm{reads}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{forwards}\:\mathrm{or}\:\mathrm{backwards}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{only}\:\mathrm{prime}\:\mathrm{palindrome}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{even}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}?\: \\ $$

Question Number 100677    Answers: 0   Comments: 0

for m,n positive integers m > n prove that lcd(m,n) + lcd(m+1,n+1) > ((2mn)/(√(m−n)))

$$\mathrm{for}\:\mathrm{m},\mathrm{n}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{m}\:>\:\mathrm{n}\: \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{lcd}\left(\mathrm{m},\mathrm{n}\right)\:+\:\mathrm{lcd}\left(\mathrm{m}+\mathrm{1},\mathrm{n}+\mathrm{1}\right)\:>\:\frac{\mathrm{2mn}}{\sqrt{\mathrm{m}−\mathrm{n}}} \\ $$

Question Number 100583    Answers: 0   Comments: 0

A transformation f on a complex plane is defined by z′ = (1 +i)z −3 + 4i show that f is a simultitude with radius r and centre Ω to be determined. Determine to the invariant point under f.

$$\:\mathrm{A}\:\mathrm{transformation}\:{f}\:\mathrm{on}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{plane} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{z}'\:=\:\left(\mathrm{1}\:+{i}\right){z}\:−\mathrm{3}\:+\:\mathrm{4}{i} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{simultitude}\:\mathrm{with}\:\mathrm{radius}\:{r}\:\mathrm{and}\:\mathrm{centre} \\ $$$$\Omega\:\mathrm{to}\:\mathrm{be}\:\mathrm{determined}. \\ $$$$\mathrm{Determine}\:\mathrm{to}\:\mathrm{the}\:\mathrm{invariant}\:\mathrm{point}\:\mathrm{under}\:{f}. \\ $$

Question Number 100179    Answers: 2   Comments: 0

Question Number 100178    Answers: 2   Comments: 0

what is the number of ordered pairs of positif integers (x,y) that satisfy x^2 +y^2 −xy=37

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{positif}\: \\ $$$$\mathrm{integers}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{that}\:\mathrm{satisfy}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{xy}=\mathrm{37} \\ $$

Question Number 100042    Answers: 1   Comments: 0

Question Number 99804    Answers: 0   Comments: 3

Determine x,y ∈ Z such that 1+2^x +2^(2x+1) = y^2

$${Determine}\:{x},{y}\:\in\:\mathbb{Z}\:{such}\:{that}\: \\ $$$$\mathrm{1}+\mathrm{2}^{{x}} \:+\mathrm{2}^{\mathrm{2}{x}+\mathrm{1}} \:=\:{y}^{\mathrm{2}} \: \\ $$

Question Number 99495    Answers: 2   Comments: 2

solve for x,y ∈ N 7^y +2 = 3^x

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x},\mathrm{y}\:\in\:\mathbb{N}\: \\ $$$$\mathrm{7}^{\mathrm{y}} +\mathrm{2}\:=\:\mathrm{3}^{\mathrm{x}} \: \\ $$

Question Number 99322    Answers: 1   Comments: 0

what is remainder of 2^7^(2002) divided by 352

$${what}\:{is}\:{remainder}\:{of}\:\mathrm{2}^{\mathrm{7}^{\mathrm{2002}} } \:{divided} \\ $$$${by}\:\mathrm{352}\: \\ $$

Question Number 98602    Answers: 2   Comments: 6

find integral solution of y^2 = x^3 +1

$$\mathrm{find}\:\mathrm{integral}\:\mathrm{solution} \\ $$$$\mathrm{of}\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{x}^{\mathrm{3}} +\mathrm{1}\: \\ $$

Question Number 98596    Answers: 1   Comments: 2

Question Number 98091    Answers: 1   Comments: 0

what is number of positive integral solutions of 10xy+7x+3y = 2077829313

$$\mathrm{what}\:\mathrm{is}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integral}\: \\ $$$$\mathrm{solutions}\:\mathrm{of}\:\mathrm{10xy}+\mathrm{7x}+\mathrm{3y}\:=\:\mathrm{2077829313} \\ $$

Question Number 97823    Answers: 4   Comments: 2

If x and y are integers , prove that x^3 −7x divisible by 3

$$\mathrm{If}\:{x}\:\mathrm{and}\:{y}\:\mathrm{are}\:\mathrm{integers}\:,\:\mathrm{prove} \\ $$$$\mathrm{that}\:{x}^{\mathrm{3}} −\mathrm{7}{x}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}\: \\ $$

Question Number 97746    Answers: 1   Comments: 1

Determine all pairs (x,y) of integers satisfying 1+2^x +2^(2x+1) =y^2

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{pairs}\:\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{of}\:\mathrm{integers}\:\mathrm{satisfying}\:\mathrm{1}+\mathrm{2}^{\mathrm{x}} +\mathrm{2}^{\mathrm{2x}+\mathrm{1}} =\mathrm{y}^{\mathrm{2}} \: \\ $$

Question Number 97709    Answers: 0   Comments: 4

Question Number 97576    Answers: 2   Comments: 0

Show that RE[(1/(1−z))]=(1/2) where z = cos θ + i sinθ

$$\:\mathrm{Show}\:\mathrm{that}\:{RE}\left[\frac{\mathrm{1}}{\mathrm{1}−{z}}\right]=\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{where}\:{z}\:=\:\mathrm{cos}\:\theta\:+\:{i}\:\mathrm{sin}\theta \\ $$$$ \\ $$

Question Number 97413    Answers: 1   Comments: 0

Given that ω = e^(iθ) , θ≠ nπ , n ∈N show that (1 + ω)^n = 2^n ((1/2)θ)e^((1/2)(inθ)) please help me out on this, i′ve stumbled on it.

$$\mathrm{Given}\:\mathrm{that}\:\omega\:=\:{e}^{{i}\theta} ,\:\theta\neq\:{n}\pi\:,\:{n}\:\in\mathbb{N} \\ $$$$\mathrm{show}\:\mathrm{that}\:\left(\mathrm{1}\:+\:\omega\right)^{{n}} \:=\:\mathrm{2}^{{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right){e}^{\frac{\mathrm{1}}{\mathrm{2}}\left({in}\theta\right)} \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{out}\:\mathrm{on}\:\mathrm{this},\:\mathrm{i}'\mathrm{ve}\:\mathrm{stumbled}\:\mathrm{on}\:\mathrm{it}. \\ $$

Question Number 96928    Answers: 3   Comments: 1

69x ≡ 1 (mod 31) solve for x

$$\mathrm{69}{x}\:\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{31}\right)\: \\ $$$$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$

Question Number 96558    Answers: 1   Comments: 0

Let m and n be two positive integers satisfy (m/n) = (1/(10×12))+(1/(12×14))+(1/(14×16))+...+(1/(2012×2014)) find the smallest possible value of m+n

$$\mathrm{Let}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}\:\mathrm{be}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\: \\ $$$$\mathrm{satisfy}\:\frac{\mathrm{m}}{\mathrm{n}}\:=\:\frac{\mathrm{1}}{\mathrm{10}×\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{12}×\mathrm{14}}+\frac{\mathrm{1}}{\mathrm{14}×\mathrm{16}}+...+\frac{\mathrm{1}}{\mathrm{2012}×\mathrm{2014}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{m}+\mathrm{n}\: \\ $$

Question Number 96437    Answers: 1   Comments: 0

Question Number 95966    Answers: 1   Comments: 0

find all pairs of integer for xy+3x−4y = 29

$$\mathrm{find}\:\mathrm{all}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{integer}\:\mathrm{for}\: \\ $$$$\mathrm{xy}+\mathrm{3x}−\mathrm{4y}\:=\:\mathrm{29}\: \\ $$

Question Number 95897    Answers: 2   Comments: 0

If x∈C . find solution of 3+i(√2) = e^(ix)

$$\mathrm{If}\:{x}\in\mathbb{C}\:.\:\mathrm{find}\:\mathrm{solution}\:\mathrm{of}\: \\ $$$$\mathrm{3}+{i}\sqrt{\mathrm{2}}\:=\:{e}^{{ix}} \: \\ $$

Question Number 95868    Answers: 1   Comments: 1

5^(10) (mod 11)=?

$$\mathrm{5}^{\mathrm{10}} \left({mod}\:\mathrm{11}\right)=? \\ $$

Question Number 95768    Answers: 0   Comments: 1

find x such that x≡3 (mod5) x≡5 (mod7) x≡7(mod11)

$$\mathrm{find}\:\mathrm{x}\:\mathrm{such}\:\mathrm{that}\: \\ $$$${x}\equiv\mathrm{3}\:\left(\mathrm{mod5}\right) \\ $$$${x}\equiv\mathrm{5}\:\left(\mathrm{mod7}\right) \\ $$$${x}\equiv\mathrm{7}\left(\mathrm{mod11}\right) \\ $$

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