Question and Answers Forum

Number TheoryQuestion and Answers: Page 9

Question Number 135566 Answers: 1 Comments: 1

Let p,q and r be the distinct roots of the polynomial x^3 −22x^2 +80x−67. There exist real number A,B and C such that (1/(s^3 −22s^2 +80s−67)) = (A/(s−p)) + (B/(s−q)) + (C/(s−r)) for all real numbers s with s ∉ {p,q,r}.What is (1/A) + (1/B) + (1/C) ? (a) 243 (b) 244 (c) 245 (d)246 (e) 247

$${Let}\:{p},{q}\:{and}\:{r}\:{be}\:{the}\:{distinct}\:{roots} \\ $$$${of}\:{the}\:{polynomial}\:{x}^{\mathrm{3}} −\mathrm{22}{x}^{\mathrm{2}} +\mathrm{80}{x}−\mathrm{67}. \\ $$$${There}\:{exist}\:{real}\:{number}\:{A},{B}\:{and} \\ $$$${C}\:{such}\:{that}\:\frac{\mathrm{1}}{{s}^{\mathrm{3}} −\mathrm{22}{s}^{\mathrm{2}} +\mathrm{80}{s}−\mathrm{67}}\:= \\ $$$$\frac{{A}}{{s}−{p}}\:+\:\frac{{B}}{{s}−{q}}\:+\:\frac{{C}}{{s}−{r}}\:{for}\:{all}\:{real}\:{numbers} \\ $$$${s}\:{with}\:{s}\:\notin\:\left\{{p},{q},{r}\right\}.{What}\:{is}\: \\ $$$$\frac{\mathrm{1}}{{A}}\:+\:\frac{\mathrm{1}}{{B}}\:+\:\frac{\mathrm{1}}{{C}}\:? \\ $$$$\left({a}\right)\:\mathrm{243}\:\:\:\:\left({b}\right)\:\mathrm{244}\:\:\:\:\left({c}\right)\:\mathrm{245}\:\:\:\left({d}\right)\mathrm{246} \\ $$$$\left({e}\right)\:\mathrm{247}\: \\ $$

Question Number 135557 Answers: 1 Comments: 0

(1/(1−cos θ−i sin θ)) =? i=(√(−1))

$$\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\:\theta−{i}\:\mathrm{sin}\:\theta}\:=? \\ $$$${i}=\sqrt{−\mathrm{1}} \\ $$

Question Number 134960 Answers: 1 Comments: 0

Number theory A palindrome is a number that reads the same backwards as forwards, as 3141413. (a)How many two-digit palindromes are there? (b)How many three-digit ones? (c)How many k-digits ones?

$$\mathrm{Number}\:\mathrm{theory} \\ $$A palindrome is a number that reads the same backwards as forwards, as 3141413. (a)How many two-digit palindromes are there? (b)How many three-digit ones? (c)How many k-digits ones?

Question Number 134458 Answers: 0 Comments: 0

Find the last two digits of 2025^(2052) + 1392^(1329) ?

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\: \\ $$$$\:\mathrm{2025}^{\mathrm{2052}} \:+\:\mathrm{1392}^{\mathrm{1329}} \:? \\ $$

Question Number 134393 Answers: 0 Comments: 0

Σ_(k=0) ^∞ ((4^k (k!)^2 )/((2k+1)^2 (2k)!)) =?

$$\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{4}^{\mathrm{k}} \:\left(\mathrm{k}!\right)^{\mathrm{2}} }{\left(\mathrm{2k}+\mathrm{1}\right)^{\mathrm{2}} \:\left(\mathrm{2k}\right)!}\:=? \\ $$

Question Number 134327 Answers: 1 Comments: 0

Question Number 134267 Answers: 1 Comments: 0

{ ((x≡ 4 (mod 5))),((x≡ 3 (mod 4 ))) :}

$$\begin{cases}{\mathrm{x}\equiv\:\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{5}\right)}\\{\mathrm{x}\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{4}\:\right)}\end{cases} \\ $$

Question Number 134252 Answers: 1 Comments: 0

solve { ((x≡ 2 (mod 3))),((x≡ 5 (mod 7))) :}

$$\:\mathrm{solve}\:\begin{cases}{\mathrm{x}\equiv\:\mathrm{2}\:\left(\mathrm{mod}\:\mathrm{3}\right)}\\{\mathrm{x}\equiv\:\mathrm{5}\:\left(\mathrm{mod}\:\mathrm{7}\right)}\end{cases} \\ $$

Question Number 134208 Answers: 0 Comments: 0

Determine if the series Σ_(n=1) ^∞ a_n by the formula converges or diverges . a_1 = 7, a_(n+1) = ((9n+3sin n)/(4n+5cos n)).a_n (a) converges (b) diverges

$$\mathrm{Determine}\:\mathrm{if}\:\mathrm{the}\:\mathrm{series}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{a}_{\mathrm{n}} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{formula}\:\mathrm{converges}\:\mathrm{or}\: \\ $$$$\mathrm{diverges}\:.\:\mathrm{a}_{\mathrm{1}} =\:\mathrm{7},\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:=\:\frac{\mathrm{9n}+\mathrm{3sin}\:\mathrm{n}}{\mathrm{4n}+\mathrm{5cos}\:\mathrm{n}}.\mathrm{a}_{\mathrm{n}} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{diverges} \\ $$$$ \\ $$

Question Number 134069 Answers: 1 Comments: 2

Given a,b and c are real numbers and a<b<c. If (1/a) + (1/b) + (1/c) = (1/(18)) , find minimum value of a.

$${Given}\:{a},{b}\:{and}\:{c}\:{are}\:{real}\:{numbers}\:{and}\:{a}<{b}<{c}. \\ $$$${If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{\mathrm{18}}\:,\:{find}\:{minimum}\:{value}\:{of}\:{a}. \\ $$

Question Number 134037 Answers: 1 Comments: 0

Question Number 133897 Answers: 2 Comments: 0

Find the least positive integer that leaves a remainder 3 when divided by 7 , 4 when divided by 9 , and 8 when divided by 11.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{positive}\:\mathrm{integer} \\ $$$$\mathrm{that}\:\mathrm{leaves}\:\mathrm{a}\:\mathrm{remainder}\:\mathrm{3}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{7} \\ $$$$,\:\mathrm{4}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{9}\:,\:\mathrm{and}\:\mathrm{8}\:\mathrm{when} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\mathrm{11}. \\ $$

Question Number 133761 Answers: 1 Comments: 0

Solve the linear congruence 19x ≡ 4 (mod 141 )

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{linear}\:\mathrm{congruence}\: \\ $$$$\mathrm{19}{x}\:\equiv\:\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{141}\:\right) \\ $$

Question Number 133757 Answers: 1 Comments: 0

Question Number 133611 Answers: 1 Comments: 0

Given system of equation { ((2x−3y = 13)),((3x+2y = b)) :} , where l ≤ b≤ 100 and b is integer. Suppose n^2 = x+y where x,y is solution of given system of equation , find the value of n for n is integer

$$\mathrm{Given}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation}\: \\ $$$$\begin{cases}{\mathrm{2x}−\mathrm{3y}\:=\:\mathrm{13}}\\{\mathrm{3x}+\mathrm{2y}\:=\:\mathrm{b}}\end{cases}\:,\:\mathrm{where}\:\mathrm{l}\:\leqslant\:\mathrm{b}\leqslant\:\mathrm{100}\:\mathrm{and} \\ $$$$\mathrm{b}\:\mathrm{is}\:\mathrm{integer}.\:\mathrm{Suppose}\:\mathrm{n}^{\mathrm{2}} \:=\:\mathrm{x}+\mathrm{y}\:\mathrm{where} \\ $$$$\mathrm{x},\mathrm{y}\:\mathrm{is}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{given}\:\mathrm{system}\: \\ $$$$\mathrm{of}\:\mathrm{equation}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n} \\ $$$$\mathrm{for}\:\mathrm{n}\:\mathrm{is}\:\mathrm{integer}\: \\ $$

Question Number 133412 Answers: 1 Comments: 0

What are the last two digits of 2^(222) −1 ?

$$\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{2}^{\mathrm{222}} −\mathrm{1}\:? \\ $$

Question Number 133320 Answers: 1 Comments: 0

Find all n for which n^2 +2n+4 is divisible by 7

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\:\mathrm{n}^{\mathrm{2}} +\mathrm{2n}+\mathrm{4}\: \\ $$$$\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}\: \\ $$

Question Number 132386 Answers: 1 Comments: 0

Find the value of { ((ln i)),((ln (3+4i))) :}

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\begin{cases}{\mathrm{ln}\:{i}}\\{\mathrm{ln}\:\left(\mathrm{3}+\mathrm{4}{i}\right)}\end{cases} \\ $$

Question Number 131497 Answers: 0 Comments: 1

please recommend problem and exercise book for number theory where answers and solutions are not given or only given at the end of the book. i find it annoying when answers are always right next to the question. a book that has no solutions would be even greater. the only such book i have found is 250 problems in elementary number theory. thank.

$$\: \\ $$$$\: \\ $$$$\:\mathrm{please}\:\mathrm{recommend}\:\mathrm{problem}\:\mathrm{and}\:\mathrm{exercise}\:\mathrm{book}\:\mathrm{for}\:\mathrm{number}\:\mathrm{theory}\: \\ $$$$\:\mathrm{where}\:\mathrm{answers}\:\mathrm{and}\:\mathrm{solutions}\:\mathrm{are}\:\mathrm{not}\:\mathrm{given}\:\mathrm{or}\:\mathrm{only} \\ $$$$\:\mathrm{given}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{book}.\:\mathrm{i}\:\mathrm{find}\:\mathrm{it}\:\mathrm{annoying}\:\mathrm{when}\:\mathrm{answers}\:\mathrm{are}\:\mathrm{always}\:\mathrm{right}\:\mathrm{next}\:\mathrm{to}\:\mathrm{the}\:\mathrm{question}. \\ $$$$\:\mathrm{a}\:\mathrm{book}\:\mathrm{that}\:\mathrm{has}\:\mathrm{no}\:\mathrm{solutions}\:\mathrm{would}\:\mathrm{be}\:\mathrm{even}\:\mathrm{greater}. \\ $$$$\:\mathrm{the}\:\mathrm{only}\:\mathrm{such}\:\mathrm{book}\:\mathrm{i}\:\mathrm{have}\:\mathrm{found}\:\mathrm{is}\:\mathrm{250}\:\mathrm{problems}\:\mathrm{in}\:\mathrm{elementary}\:\mathrm{number}\:\mathrm{theory}. \\ $$$$\:\mathrm{thank}. \\ $$$$\: \\ $$$$\: \\ $$

Question Number 131484 Answers: 3 Comments: 0

Express Σ_(n=1) ^(49) (1/( (√(n+(√(n^2 −1)))))) as a+b(√2) for some integers a and b

$$\mathrm{Express}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{49}} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}+\sqrt{\mathrm{n}^{\mathrm{2}} −\mathrm{1}}}}\:\mathrm{as}\:{a}+{b}\sqrt{\mathrm{2}} \\ $$$$\mathrm{for}\:\mathrm{some}\:\mathrm{integers}\:{a}\:\mathrm{and}\:{b} \\ $$

Question Number 131196 Answers: 1 Comments: 0

The minimum value of the expression B = ∣z∣^2 +∣z−3∣^2 +∣z−6i∣^2 is p. What the value of (p/(10)) .?

$$\:{The}\:{minimum}\:{value}\:{of}\:{the}\: \\ $$$${expression}\:{B}\:=\:\mid{z}\mid^{\mathrm{2}} +\mid{z}−\mathrm{3}\mid^{\mathrm{2}} +\mid{z}−\mathrm{6}{i}\mid^{\mathrm{2}} \\ $$$${is}\:{p}.\:{What}\:{the}\:{value}\:{of}\:\frac{{p}}{\mathrm{10}}\:.? \\ $$

Question Number 130727 Answers: 2 Comments: 2