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Number TheoryQuestion and Answers: Page 23

Question Number 9485    Answers: 0   Comments: 2

a_n = a_(n−1) ^2 + a_(n−2) ^2 If a_1 =1 and a_2 =1, what is the remainder of a_(2016) when divided by 10 ?

$${a}_{{n}} \:=\:{a}_{{n}−\mathrm{1}} ^{\mathrm{2}} \:+\:{a}_{{n}−\mathrm{2}} ^{\mathrm{2}} \\ $$$$\mathrm{If}\:{a}_{\mathrm{1}} =\mathrm{1}\:\mathrm{and}\:{a}_{\mathrm{2}} =\mathrm{1},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{of}\:{a}_{\mathrm{2016}} \:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{10}\:? \\ $$

Question Number 9069    Answers: 0   Comments: 2

Question Number 9049    Answers: 0   Comments: 5

Prove that every even number can be expressed as sum of two primes or give an counter example.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{every}\:\mathrm{even}\:\mathrm{number}\:\mathrm{can}\:\mathrm{be}\: \\ $$$$\mathrm{expressed}\:\mathrm{as}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{primes}\:\mathrm{or} \\ $$$$\mathrm{give}\:\mathrm{an}\:\mathrm{counter}\:\mathrm{example}. \\ $$

Question Number 9025    Answers: 0   Comments: 7

Determine number/s that is/are comprised of four distinct prime factors such that difference of largest and smallest prime factors is equal to the sum of remaining two factors. _(Propsed by Rasheed Soomro)

$$\mathrm{Determine}\:\mathrm{number}/\mathrm{s}\:\mathrm{that}\:\mathrm{is}/\mathrm{are}\:\mathrm{comprised} \\ $$$$\mathrm{of}\:\mathrm{four}\:\mathrm{distinct}\:\mathrm{prime}\:\mathrm{factors}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{difference}\:\mathrm{of}\:\mathrm{largest}\:\mathrm{and}\:\mathrm{smallest}\:\mathrm{prime} \\ $$$$\mathrm{factors}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{remaining} \\ $$$$\mathrm{two}\:\mathrm{factors}.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:_{\mathrm{Propsed}\:\mathrm{by}\:\mathrm{Rasheed}\:\mathrm{Soomro}} \\ $$

Question Number 9021    Answers: 1   Comments: 0

What is the remainder when (13^5 + 14^5 + 15^5 + 16^5 ) is divided by 29 ?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\: \\ $$$$\left(\mathrm{13}^{\mathrm{5}} \:+\:\mathrm{14}^{\mathrm{5}} \:+\:\mathrm{15}^{\mathrm{5}} \:+\:\mathrm{16}^{\mathrm{5}} \right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{29}\:?\: \\ $$

Question Number 8838    Answers: 1   Comments: 0

Show that : e^(iπ + 1) = 0

$$\mathrm{Show}\:\mathrm{that}\::\:\:\mathrm{e}^{\mathrm{i}\pi\:+\:\mathrm{1}} \:=\:\mathrm{0} \\ $$

Question Number 8347    Answers: 0   Comments: 5

a_1 =2 , a_(n+1) >a_n (a_(n+1) −a_n )^2 = 2(a_(n+1) +a_n ) a_n =?? help me please.

$${a}_{\mathrm{1}} =\mathrm{2}\:,\:\:{a}_{{n}+\mathrm{1}} >{a}_{{n}} \\ $$$$\left({a}_{{n}+\mathrm{1}} −{a}_{{n}} \right)^{\mathrm{2}} =\:\mathrm{2}\left({a}_{{n}+\mathrm{1}} +{a}_{{n}} \right) \\ $$$$\:{a}_{{n}} =?? \\ $$$${help}\:{me}\:{please}. \\ $$

Question Number 8336    Answers: 0   Comments: 2

Determine smallest n(≠0), for which (ω+i)^n =1.

$$\mathrm{Determine}\:\mathrm{smallest}\:\mathrm{n}\left(\neq\mathrm{0}\right),\:\mathrm{for}\:\mathrm{which} \\ $$$$\left(\omega+\mathrm{i}\right)^{\mathrm{n}} =\mathrm{1}. \\ $$

Question Number 8302    Answers: 0   Comments: 0

find all possible values of x and y satisfying 1! + 2! + 3! + ... + x! = y^2

$$\mathrm{find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{satisfying}\: \\ $$$$\mathrm{1}!\:+\:\mathrm{2}!\:+\:\mathrm{3}!\:+\:...\:+\:\mathrm{x}!\:=\:\mathrm{y}^{\mathrm{2}} \\ $$

Question Number 8174    Answers: 0   Comments: 0

Prove that there are infinite prime numbers of the form 10^n +1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{infinite}\:\mathrm{prime} \\ $$$$\mathrm{numbers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{10}^{{n}} +\mathrm{1} \\ $$

Question Number 8139    Answers: 1   Comments: 4

is it correct? when S_n ={ 1×2+1×3+1×4+1×5+.......+1×n +2×3+2×4+2×5+.......+2×n +3×4+3×5+.......+3×n +4×5+.......+4×n .... +(n−1)×n } find S_n . /////////////// S_n +S_n +(1^2 +2^2 +3^2 +4^2 +...+n^2 )={ 1×1+1×2+1×3+1×4+.......+1×n+ 2×1+2×2+2×3+2×4+.......+2×n+ 3×1+3×2+3×3+3×4+.......+3×n+ 4×1+4×1+4×3+4×4+.......+4×n+ ... n×1+n×2+n×3+n×4+......+n×n } ⇔ =(1+2+3+4+...+n)(1+2+3+4+...+n) ⇔ 2S_n +((n(n+1)(2n+1))/6)={((n(n+1))/2)}^2 2S_n =((n(n+1))/2){((n(n+1))/2)−((2n+1)/3)} =((n(n+1))/2)×((3n^2 −n−2)/6) S_n =((n(n+1))/4)×(((3n+2)(n−1))/6) S_n =(((n−1)n(n+1)(3n+2))/(24))

$${is}\:{it}\:{correct}? \\ $$$${when} \\ $$$${S}_{{n}} =\left\{\right. \\ $$$$\mathrm{1}×\mathrm{2}+\mathrm{1}×\mathrm{3}+\mathrm{1}×\mathrm{4}+\mathrm{1}×\mathrm{5}+.......+\mathrm{1}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:+\mathrm{2}×\mathrm{3}+\mathrm{2}×\mathrm{4}+\mathrm{2}×\mathrm{5}+.......+\mathrm{2}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{3}×\mathrm{4}+\mathrm{3}×\mathrm{5}+.......+\mathrm{3}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{4}×\mathrm{5}+.......+\mathrm{4}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\left({n}−\mathrm{1}\right)×{n} \\ $$$$\left.\right\} \\ $$$${find}\:{S}_{{n}} \:. \\ $$$$/////////////// \\ $$$${S}_{{n}} +{S}_{{n}} +\left(\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} +...+{n}^{\mathrm{2}} \right)=\left\{\right. \\ $$$$\mathrm{1}×\mathrm{1}+\mathrm{1}×\mathrm{2}+\mathrm{1}×\mathrm{3}+\mathrm{1}×\mathrm{4}+.......+\mathrm{1}×{n}+ \\ $$$$\mathrm{2}×\mathrm{1}+\mathrm{2}×\mathrm{2}+\mathrm{2}×\mathrm{3}+\mathrm{2}×\mathrm{4}+.......+\mathrm{2}×{n}+ \\ $$$$\mathrm{3}×\mathrm{1}+\mathrm{3}×\mathrm{2}+\mathrm{3}×\mathrm{3}+\mathrm{3}×\mathrm{4}+.......+\mathrm{3}×{n}+ \\ $$$$\mathrm{4}×\mathrm{1}+\mathrm{4}×\mathrm{1}+\mathrm{4}×\mathrm{3}+\mathrm{4}×\mathrm{4}+.......+\mathrm{4}×{n}+ \\ $$$$... \\ $$$$\left.{n}×\mathrm{1}+{n}×\mathrm{2}+{n}×\mathrm{3}+{n}×\mathrm{4}+......+{n}×{n}\:\right\} \\ $$$$\Leftrightarrow \\ $$$$=\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+...+{n}\right)\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+...+{n}\right) \\ $$$$\Leftrightarrow \\ $$$$\mathrm{2}{S}_{{n}} +\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{6}}=\left\{\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\right\}^{\mathrm{2}} \\ $$$$\mathrm{2}{S}_{{n}} =\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\left\{\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}−\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{3}}\right\} \\ $$$$=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}×\frac{\mathrm{3}{n}^{\mathrm{2}} −{n}−\mathrm{2}}{\mathrm{6}} \\ $$$${S}_{{n}} =\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{4}}×\frac{\left(\mathrm{3}{n}+\mathrm{2}\right)\left({n}−\mathrm{1}\right)}{\mathrm{6}} \\ $$$${S}_{{n}} =\frac{\left({n}−\mathrm{1}\right){n}\left({n}+\mathrm{1}\right)\left(\mathrm{3}{n}+\mathrm{2}\right)}{\mathrm{24}} \\ $$$$ \\ $$

Question Number 7859    Answers: 1   Comments: 0

Find the remainder if 49^(1296) × 7^(131) is divided by 13

$${Find}\:{the}\:{remainder}\:{if}\:\:\:\mathrm{49}^{\mathrm{1296}} \:×\:\mathrm{7}^{\mathrm{131}} \:\:{is}\:{divided} \\ $$$${by}\:\:\mathrm{13}\:\: \\ $$

Question Number 7748    Answers: 1   Comments: 0

Given that Z and H are complex number. obtain the real and imaginary of Z^H

$${Given}\:{that}\:{Z}\:{and}\:{H}\:{are}\:{complex}\:{number}.\: \\ $$$${obtain}\:{the}\:{real}\:{and}\:{imaginary}\:{of}\:{Z}^{{H}} \\ $$

Question Number 7249    Answers: 0   Comments: 11

w,x,y,z are digits in respective base system and a,b are bases. Find out an example/examples which satisfy the following wxyz_a +wxyz_b =wxyz_(a+b)

$${w},{x},{y},{z}\:{are}\:{digits}\:{in}\:{respective}\:{base}\:{system}\:{and} \\ $$$${a},{b}\:{are}\:{bases}. \\ $$$${Find}\:{out}\:{an}\:{example}/{examples}\:{which}\:{satisfy} \\ $$$${the}\:{following} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{wxyz}_{{a}} +{wxyz}_{{b}} ={wxyz}_{{a}+{b}} \: \\ $$

Question Number 7213    Answers: 0   Comments: 0

Let a and b be positive integers such that ab+1 divides a^2 +b^2 . Show that ((a^2 +b^2 )/(ab+1)) is the square of an integer. (IMO 1988 Qu.6)

$${Let}\:{a}\:{and}\:{b}\:{be}\:{positive}\:{integers}\:{such} \\ $$$${that}\:{ab}+\mathrm{1}\:{divides}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} .\:{Show}\:{that} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{{ab}+\mathrm{1}}\:{is}\:{the}\:{square}\:{of}\:{an}\:{integer}. \\ $$$$\left({IMO}\:\mathrm{1988}\:{Qu}.\mathrm{6}\right) \\ $$

Question Number 7230    Answers: 0   Comments: 2

An Interesting App: I′ve recently downloaded the app called ′Math Tricks′ from the Google Play Store. If you′d like to improve your speed and skill in the mental calculations arena, I think this app should be of interest to you. The app can throw you simple calculations that include the basic operations of addition,subtraction,division and multiplication. The application can show you tricks to calculating 52^2 or 125^2 ,for example, under 10 seconds mentally. So, check it out! I′ve seen that the levels of difficulty on the app can go as high as mentally calculating positive integers raised to the power of 9, and also finding 9th roots of positive integers! Yozzia

$${An}\:{Interesting}\:{App}: \\ $$$${I}'{ve}\:{recently}\:{downloaded}\:{the}\:{app}\:{called} \\ $$$$'{Math}\:{Tricks}'\:{from}\:{the}\:{Google}\:{Play}\:{Store}. \\ $$$${If}\:{you}'{d}\:{like}\:{to}\:{improve}\:{your}\:{speed} \\ $$$${and}\:{skill}\:{in}\:{the}\:{mental}\:{calculations} \\ $$$${arena},\:{I}\:{think}\:{this}\:{app}\:{should}\:{be}\:{of} \\ $$$${interest}\:{to}\:{you}.\:{The}\:{app}\:{can}\:{throw}\:{you}\:{simple} \\ $$$${calculations}\:{that}\:{include}\:{the}\:{basic}\:{operations} \\ $$$${of}\:{addition},{subtraction},{division}\:{and} \\ $$$${multiplication}.\:{The}\:{application}\:{can} \\ $$$${show}\:{you}\:{tricks}\:{to}\:{calculating}\:\mathrm{52}^{\mathrm{2}} \:{or} \\ $$$$\mathrm{125}^{\mathrm{2}} ,{for}\:{example},\:{under}\:\mathrm{10}\:{seconds} \\ $$$${mentally}.\:{So},\:{check}\:{it}\:{out}!\:{I}'{ve}\:{seen} \\ $$$${that}\:{the}\:{levels}\:{of}\:{difficulty}\:{on}\:{the}\: \\ $$$${app}\:{can}\:{go}\:{as}\:{high}\:{as}\:{mentally}\:{calculating}\:{positive} \\ $$$${integers}\:{raised}\:{to}\:{the}\:{power}\:{of}\:\mathrm{9}, \\ $$$${and}\:{also}\:{finding}\:\mathrm{9}{th}\:{roots}\:{of}\: \\ $$$${positive}\:{integers}! \\ $$$$ \\ $$$${Yozzia} \\ $$

Question Number 7191    Answers: 1   Comments: 0

Evaluate Σ ((sin(3n))/n) from 1 to infinity

$${Evaluate}\:\:\:\:\:\Sigma\:\frac{{sin}\left(\mathrm{3}{n}\right)}{{n}}\:\:\:\:\:{from}\:\:\mathrm{1}\:\:{to}\:\:{infinity}\: \\ $$

Question Number 7189    Answers: 0   Comments: 2

Evaluate : Σ ((sin(n))/n) , From 1 to infinity.

$${Evaluate}\:\:\::\:\:\:\Sigma\:\frac{{sin}\left({n}\right)}{{n}}\:\:,\:\:\:{From}\:\:\:\mathrm{1}\:{to}\:\:{infinity}. \\ $$

Question Number 7005    Answers: 0   Comments: 0

Is u_n real sequence defende by u_0 = 3e^t u_(n+1) = 2(u_n )^2 − 1 Determine the general term u_n of this series (justify your answer and method used)

$${Is}\:\:{u}_{{n}} \:{real}\:{sequence}\:{defende}\:\:{by}\:\: \\ $$$${u}_{\mathrm{0}} \:=\:\mathrm{3}{e}^{{t}} \:{u}_{{n}+\mathrm{1}} \:=\:\mathrm{2}\left({u}_{{n}} \right)^{\mathrm{2}} \:−\:\mathrm{1} \\ $$$${Determine}\:{the}\:{general}\:{term}\:{u}_{{n}} \:{of}\:{this}\:{series}\: \\ $$$$\left({justify}\:{your}\:{answer}\:{and}\:{method}\:{used}\right) \\ $$$$ \\ $$

Question Number 6280    Answers: 1   Comments: 1

52 : 9 :: 48 : 31 :: 27 : 13 :: 65 : ?

$$\mathrm{52}\::\:\mathrm{9}\:::\:\mathrm{48}\::\:\mathrm{31}\:::\:\mathrm{27}\::\:\mathrm{13}\:::\:\mathrm{65}\::\:? \\ $$

Question Number 6055    Answers: 1   Comments: 0

ln(x)+x=a x=?

$${ln}\left({x}\right)+{x}={a} \\ $$$${x}=? \\ $$

Question Number 5947    Answers: 0   Comments: 0

Show that 3!^(5!^(7!^(9!^(...2013!) ) ) ) ≡1 (mod 11).

$$ \\ $$$${Show}\:{that}\:\mathrm{3}!^{\mathrm{5}!^{\mathrm{7}!^{\mathrm{9}!^{...\mathrm{2013}!} } } } \:\equiv\mathrm{1}\:\left({mod}\:\mathrm{11}\right). \\ $$

Question Number 5796    Answers: 0   Comments: 6

How many ways can you express 30,030 as the product of 4 positive numbers? (excluding 1)

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{you}\:\mathrm{express} \\ $$$$\mathrm{30},\mathrm{030}\:\mathrm{as}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{4}\:\mathrm{positive}\:\mathrm{numbers}? \\ $$$$\left(\mathrm{excluding}\:\mathrm{1}\right) \\ $$

Question Number 5749    Answers: 0   Comments: 2

if a≡b(mod c) does: b≡a(mod c) ???

$$\mathrm{if}\:{a}\equiv{b}\left(\mathrm{mod}\:{c}\right) \\ $$$${does}: \\ $$$${b}\equiv{a}\left(\mathrm{mod}\:{c}\right) \\ $$$$??? \\ $$

Question Number 5216    Answers: 1   Comments: 0

Let p_j represent the j−th prime number. Now, define the number n whose decimal representation is written out in terms of p_j (j∈N) in the following way: n=0.p_1 p_2 p_3 p_4 p_5 ...p_j p_(j+1) p_(j+2) ... or n=0.(2)(3)(5)(7)(11)...(521)(523)(541)... ⇒n=0.235711...521523541... Prove or disprove that n is irrational.

$${Let}\:{p}_{{j}} \:{represent}\:{the}\:{j}−{th}\:{prime}\:{number}. \\ $$$${Now},\:{define}\:{the}\:{number}\:{n}\:{whose} \\ $$$${decimal}\:{representation}\:{is}\:{written}\:{out} \\ $$$${in}\:{terms}\:{of}\:{p}_{{j}} \:\left({j}\in\mathbb{N}\right)\:{in}\:{the}\:{following} \\ $$$${way}: \\ $$$${n}=\mathrm{0}.{p}_{\mathrm{1}} {p}_{\mathrm{2}} {p}_{\mathrm{3}} {p}_{\mathrm{4}} {p}_{\mathrm{5}} ...{p}_{{j}} {p}_{{j}+\mathrm{1}} {p}_{{j}+\mathrm{2}} ... \\ $$$${or}\:{n}=\mathrm{0}.\left(\mathrm{2}\right)\left(\mathrm{3}\right)\left(\mathrm{5}\right)\left(\mathrm{7}\right)\left(\mathrm{11}\right)...\left(\mathrm{521}\right)\left(\mathrm{523}\right)\left(\mathrm{541}\right)... \\ $$$$\Rightarrow{n}=\mathrm{0}.\mathrm{235711}...\mathrm{521523541}... \\ $$$${Prove}\:{or}\:{disprove}\:{that}\:{n}\:{is}\:{irrational}. \\ $$$$ \\ $$$$ \\ $$

Question Number 5185    Answers: 1   Comments: 0

If 2^(x ) and 3^x are integers for some x∈R^+ , must x be an integer?

$${If}\:\:\mathrm{2}^{{x}\:} \:{and}\:\:\mathrm{3}^{{x}} \:{are}\:{integers}\:{for}\:{some} \\ $$$${x}\in\mathbb{R}^{+} ,\:{must}\:{x}\:{be}\:{an}\:{integer}?\: \\ $$

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