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Number TheoryQuestion and Answers: Page 7

Question Number 140833 Answers: 3 Comments: 1

Determine if the series Σ_(n=1) ^∞ a_n defined by the formula converges or diverges . a_1 = 4 , a_(n+1) = ((10+sin n)/n). a_n

$$\mathrm{Determine}\:\mathrm{if}\:\mathrm{the}\:\mathrm{series}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{a}_{\mathrm{n}} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\mathrm{the}\:\:\mathrm{formula}\:\mathrm{converges}\:\mathrm{or} \\ $$$$\mathrm{diverges}\:.\:\mathrm{a}_{\mathrm{1}} =\:\mathrm{4}\:,\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:=\:\frac{\mathrm{10}+\mathrm{sin}\:\mathrm{n}}{\mathrm{n}}.\:\mathrm{a}_{\mathrm{n}} \\ $$

Question Number 140428 Answers: 1 Comments: 1

If 4x=3(Mod 6), find the first four values of x.

$$\mathrm{If}\:\mathrm{4}{x}=\mathrm{3}\left(\mathrm{Mod}\:\mathrm{6}\right),\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{four}\:\mathrm{values}\:\mathrm{of}\:{x}. \\ $$

Question Number 140329 Answers: 2 Comments: 0

x^(⌊x⌋) + x^(⌈x⌉) = ((175)/8)

$$\:\mathrm{x}^{\lfloor\mathrm{x}\rfloor} \:+\:\mathrm{x}^{\lceil\mathrm{x}\rceil} \:=\:\frac{\mathrm{175}}{\mathrm{8}} \\ $$

Question Number 140449 Answers: 1 Comments: 0

Find the first four values of 2x+5=1(mod 7)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{four}\:\mathrm{values}\:\mathrm{of} \\ $$$$\mathrm{2}{x}+\mathrm{5}=\mathrm{1}\left(\mathrm{mod}\:\mathrm{7}\right) \\ $$

Question Number 139668 Answers: 1 Comments: 3

Question Number 139560 Answers: 0 Comments: 0

Σ_(n_1 +2n_2 +3n_3 +..+rn_r =n) ^n (1/(n_1 !n_2 !n_3 !..n_r !1^n_1 2^n_2 3^n_3 4^n_4 ...r^n_r ))=1 0≥n_1 ,n_2 ,n_3 ,..≥n Prove the above identity

$$\underset{{n}_{\mathrm{1}} +\mathrm{2}{n}_{\mathrm{2}} +\mathrm{3}{n}_{\mathrm{3}} +..+{rn}_{{r}} ={n}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}_{\mathrm{1}} !{n}_{\mathrm{2}} !{n}_{\mathrm{3}} !..{n}_{{r}} !\mathrm{1}^{{n}_{\mathrm{1}} } \mathrm{2}^{{n}_{\mathrm{2}} } \mathrm{3}^{{n}_{\mathrm{3}} } \mathrm{4}^{{n}_{\mathrm{4}} } ...{r}^{{n}_{{r}} } }=\mathrm{1} \\ $$$$\mathrm{0}\geqslant{n}_{\mathrm{1}} ,{n}_{\mathrm{2}} ,{n}_{\mathrm{3}} ,..\geqslant{n} \\ $$$$\mathrm{P}{rove}\:\mathrm{t}{he}\:{above}\:{identity} \\ $$

Question Number 138003 Answers: 2 Comments: 0

Question Number 137635 Answers: 0 Comments: 1

Question Number 137585 Answers: 2 Comments: 0

Find the cube of the number N= (√(7(√(3(√(7(√(3(√(7(√(3(√(7(√(3...))))))))))))))))

$${Find}\:{the}\:{cube}\:{of}\:{the}\:{number}\: \\ $$$${N}=\:\sqrt{\mathrm{7}\sqrt{\mathrm{3}\sqrt{\mathrm{7}\sqrt{\mathrm{3}\sqrt{\mathrm{7}\sqrt{\mathrm{3}\sqrt{\mathrm{7}\sqrt{\mathrm{3}...}}}}}}}} \\ $$

Question Number 137579 Answers: 1 Comments: 0

(−1)×(1/(π.i)) =?

$$\left(−\mathrm{1}\right)×\frac{\mathrm{1}}{\pi.{i}}\:=?\: \\ $$

Question Number 137383 Answers: 3 Comments: 0

What is the remainder 13^(163) when divided by 99

$${What}\:{is}\:{the}\:{remainder}\:\mathrm{13}^{\mathrm{163}} \:{when} \\ $$$${divided}\:{by}\:\mathrm{99}\: \\ $$

Question Number 137364 Answers: 3 Comments: 0

Find the remainder 7^(30) divide by 10

$${Find}\:{the}\:{remainder}\:\mathrm{7}^{\mathrm{30}} \:{divide} \\ $$$${by}\:\mathrm{10}\: \\ $$

Question Number 136996 Answers: 0 Comments: 0

Given a,b and c is real number satisfy a+b+c = 4 and ab+ac+bc = 3 . The value of ⌈ 3c+2 ⌉ = ?

$$\mathrm{Given}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{is}\:\mathrm{real}\:\mathrm{number}\:\mathrm{satisfy} \\ $$$$\mathrm{a}+\mathrm{b}+\mathrm{c}\:=\:\mathrm{4}\:\mathrm{and}\:\mathrm{ab}+\mathrm{ac}+\mathrm{bc}\:=\:\mathrm{3}\:.\:\mathrm{The}\:\mathrm{value} \\ $$$$\mathrm{of}\:\lceil\:\mathrm{3c}+\mathrm{2}\:\rceil\:=\:? \\ $$

Question Number 136830 Answers: 1 Comments: 0

Determine all solutions in the integers of the following Diophantine equations: (a)56x+72y=40 (b)24x+138y=18 (c)221x+35y=11

$$ \\ $$Determine all solutions in the integers of the following Diophantine equations: (a)56x+72y=40 (b)24x+138y=18 (c)221x+35y=11

Question Number 136792 Answers: 1 Comments: 0

How do you solve the diophantine equation (1)3xy +2x +y = 12 ? (2) x^3 = 4y^2 +4y−3 ?

$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{diophantine}\:\mathrm{equation} \\ $$$$\left(\mathrm{1}\right)\mathrm{3xy}\:+\mathrm{2x}\:+\mathrm{y}\:=\:\mathrm{12}\:? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{x}^{\mathrm{3}} =\:\mathrm{4y}^{\mathrm{2}} +\mathrm{4y}−\mathrm{3}\:? \\ $$

Question Number 136038 Answers: 0 Comments: 0

Find a_(n ) if (1/z^m )×coth (πz)×cot (zπ)=Σ_(n=0) ^∞ a_n z^n around z=0

$${Find}\:{a}_{{n}\:} {if} \\ $$$$\frac{\mathrm{1}}{{z}^{{m}} }×\mathrm{coth}\:\left(\pi{z}\right)×\mathrm{cot}\:\left({z}\pi\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{a}_{{n}} {z}^{{n}} \\ $$$${around}\:{z}=\mathrm{0} \\ $$

Question Number 135963 Answers: 0 Comments: 0

how many of the first triangular number have the ones zero

$$ \\ $$how many of the first triangular number have the ones zero

Question Number 135958 Answers: 0 Comments: 0

Find the value of n such that ((n(n+1))/2) = k (mod 10) for n is integer number

$${Find}\:{the}\:{value}\:{of}\:{n}\:{such}\:{that} \\ $$$$\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:=\:{k}\:\left({mod}\:\mathrm{10}\right)\:{for} \\ $$$${n}\:{is}\:{integer}\:{number}\: \\ $$

Question Number 135718 Answers: 1 Comments: 0

Question Number 135692 Answers: 1 Comments: 0

Solve the system of congruences 2x≡1(mod5) 3x≡2(mod7) 4x≡1(mod11)

$$ \\ $$Solve the system of congruences 2x≡1(mod5) 3x≡2(mod7) 4x≡1(mod11)

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

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