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Question Number 117369 Answers: 1 Comments: 0

Find the units digit of 2013^1 +2013^2 +2013^3 +...+2013^(2013)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{units}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{2013}^{\mathrm{1}} +\mathrm{2013}^{\mathrm{2}} +\mathrm{2013}^{\mathrm{3}} +...+\mathrm{2013}^{\mathrm{2013}} \\ $$

Question Number 117045 Answers: 4 Comments: 0

If x is a complex number satisfying x^2 +x+1 = 0 , what is the value of x^(53) +x^(52) +x^(51) +x^(50) +x^(49) ?

$$\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{satisfying}\: \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\:=\:\mathrm{0}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{x}^{\mathrm{53}} +\mathrm{x}^{\mathrm{52}} +\mathrm{x}^{\mathrm{51}} +\mathrm{x}^{\mathrm{50}} +\mathrm{x}^{\mathrm{49}} \:? \\ $$

Question Number 116700 Answers: 1 Comments: 0

How many positive integral solutions does 3x+5y=300 have?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{positive}\:\mathrm{integral}\: \\ $$$$\mathrm{solutions}\:\mathrm{does}\:\mathrm{3x}+\mathrm{5y}=\mathrm{300}\:\mathrm{have}? \\ $$

Question Number 116725 Answers: 4 Comments: 3

what the value of log _(10) (−1) in complex number

$$\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{log}\:_{\mathrm{10}} \left(−\mathrm{1}\right)\:\mathrm{in}\: \\ $$$$\mathrm{complex}\:\mathrm{number} \\ $$

Question Number 116356 Answers: 0 Comments: 0

Let S ={1,2,3,4,...,48,49} .What is the maximum value of n such that it is possible to select n numbers from S and arrange them in a circle in such a way that the product of any two adjacent numbers in the circle is less than 100?

$$\mathrm{Let}\:\mathrm{S}\:=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},...,\mathrm{48},\mathrm{49}\right\}\:.\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{select}\:\mathrm{n}\:\mathrm{numbers}\: \\ $$$$\mathrm{from}\:\mathrm{S}\:\mathrm{and}\:\mathrm{arrange}\:\mathrm{them}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\: \\ $$$$\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{adjacent}\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{100}? \\ $$

Question Number 115742 Answers: 1 Comments: 0

what the cooefficient of x^(10) from (1+x)×(1+2x^2 )×(1+3x^3 )×...×(1+2020x^(2020) )

$${what}\:{the}\:{cooefficient}\:{of}\:{x}^{\mathrm{10}} \:{from} \\ $$$$\left(\mathrm{1}+{x}\right)×\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)×\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{3}} \right)×...×\left(\mathrm{1}+\mathrm{2020}{x}^{\mathrm{2020}} \right) \\ $$

Question Number 115033 Answers: 2 Comments: 0

If a and b positive real number where a^(505) + b^(505) = 1, then minimum value a^(2020) + b^(2020) is __

$${If}\:{a}\:{and}\:{b}\:{positive}\:{real}\:{number}\:{where} \\ $$$${a}^{\mathrm{505}} \:+\:{b}^{\mathrm{505}} \:=\:\mathrm{1},\:{then}\:{minimum}\:{value} \\ $$$${a}^{\mathrm{2020}} \:+\:{b}^{\mathrm{2020}} \:{is}\:\_\_ \\ $$

Question Number 114401 Answers: 1 Comments: 0

What is reminder when 4^(29) divided by 17

$${What}\:{is}\:{reminder}\:{when}\:\mathrm{4}^{\mathrm{29}} \\ $$$${divided}\:{by}\:\mathrm{17} \\ $$

Question Number 114130 Answers: 1 Comments: 5

Question Number 113221 Answers: 0 Comments: 3

Does anyone know a good website for nested radicals? ((7((20))^(1/3) −19))^(1/6) =((5/3))^(1/3) −((2/3))^(1/3)

$${Does}\:{anyone}\:{know}\:{a}\:{good} \\ $$$${website}\:{for}\:{nested}\:{radicals}? \\ $$$$\sqrt[{\mathrm{6}}]{\mathrm{7}\sqrt[{\mathrm{3}}]{\mathrm{20}}−\mathrm{19}}=\sqrt[{\mathrm{3}}]{\mathrm{5}/\mathrm{3}}−\sqrt[{\mathrm{3}}]{\mathrm{2}/\mathrm{3}} \\ $$

Question Number 113199 Answers: 1 Comments: 8

Change the following decimal number into binary number: 73.108

$${Change}\:{the}\:{following}\:{decimal} \\ $$$${number}\:{into}\:{binary}\:{number}: \\ $$$$\mathrm{73}.\mathrm{108} \\ $$

Question Number 113187 Answers: 1 Comments: 0

a,b,c ∈N such that ((a(√3) +b)/(b(√3)+c)) ∈ Q, show that ((a^2 +b^2 +c^2 )/(a+b+c)) ∈ Z

$$\mathrm{a},\mathrm{b},\mathrm{c}\:\in\mathbb{N}\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{a}\sqrt{\mathrm{3}}\:+\mathrm{b}}{\mathrm{b}\sqrt{\mathrm{3}}+\mathrm{c}}\:\in\:\mathrm{Q},\:\mathrm{show} \\ $$$$\mathrm{that}\:\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} }{\mathrm{a}+\mathrm{b}+\mathrm{c}}\:\in\:\mathbb{Z} \\ $$

Question Number 113134 Answers: 3 Comments: 2

any one can explain me how to change decimal number to biner number. i′m forgot. example (315)_(10) = (...)_2 thank you

$$\mathrm{any}\:\mathrm{one}\:\mathrm{can}\:\mathrm{explain}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to} \\ $$$$\mathrm{change}\:\mathrm{decimal}\:\mathrm{number}\:\mathrm{to}\: \\ $$$$\mathrm{biner}\:\mathrm{number}.\:\mathrm{i}'\mathrm{m}\:\mathrm{forgot}. \\ $$$$\mathrm{example}\:\left(\mathrm{315}\right)_{\mathrm{10}} \:=\:\left(...\right)_{\mathrm{2}} \\ $$$$\mathrm{thank}\:\mathrm{you} \\ $$

Question Number 113101 Answers: 0 Comments: 2

Prove that GCD ((a,b),b)=(a,b)

$${Prove}\:{that}\:{GCD}\:\left(\left({a},{b}\right),{b}\right)=\left({a},{b}\right) \\ $$

Question Number 113091 Answers: 1 Comments: 0

What is the area bounded by the curves arg(z) = (π/3) ; arg(z)= ((2π)/3) and arg(z−2−2i(√3))=π on the complex plane?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curves} \\ $$$$\mathrm{arg}\left(\mathrm{z}\right)\:=\:\frac{\pi}{\mathrm{3}}\:;\:\mathrm{arg}\left(\mathrm{z}\right)=\:\frac{\mathrm{2}\pi}{\mathrm{3}}\:\mathrm{and}\:\mathrm{arg}\left(\mathrm{z}−\mathrm{2}−\mathrm{2i}\sqrt{\mathrm{3}}\right)=\pi \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}? \\ $$

Question Number 113073 Answers: 1 Comments: 0

Find all positive integers n for which 5^n +1 is divisible by 7

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}\:\mathrm{for}\: \\ $$$$\mathrm{which}\:\mathrm{5}^{\mathrm{n}} +\mathrm{1}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7} \\ $$

Question Number 113006 Answers: 1 Comments: 0

a+(1/(b+(1/(c+(1/(d+...))))))=(2)^(1/3) If a,b,c,d,... are positive integers, then what is the value of ′b′?

$$\mathrm{a}+\frac{\mathrm{1}}{\mathrm{b}+\frac{\mathrm{1}}{\mathrm{c}+\frac{\mathrm{1}}{\mathrm{d}+...}}}=\sqrt[{\mathrm{3}}]{\mathrm{2}} \\ $$$$\mathrm{If}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},...\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:'\mathrm{b}'? \\ $$

Question Number 113005 Answers: 1 Comments: 0

Two different two−digit natural numbers are written beside each other such that the larger number is written on the left. When the absolute difference of the two numbers is subtracted from the four−digit number so formed, the number obtained is 5481. What is the sum of the two−digit numbers?

$$\mathrm{Two}\:\mathrm{different}\:\mathrm{two}−\mathrm{digit}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{written}\:\mathrm{beside}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{larger}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{written}\:\mathrm{on}\:\mathrm{the}\:\mathrm{left}.\:\mathrm{When}\:\mathrm{the} \\ $$$$\mathrm{absolute}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:\mathrm{is}\:\mathrm{subtracted}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{number}\:\mathrm{so}\:\mathrm{formed},\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{obtained}\:\mathrm{is}\:\mathrm{5481}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}−\mathrm{digit}\:\mathrm{numbers}? \\ $$

Question Number 113003 Answers: 1 Comments: 2

The digits of a three−digit number A are written in the reverse order to form another three−digit number B. If B>A and B−A is perfectly divisible by 7. Find the range of values of A.

$$\mathrm{The}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{a}\:\mathrm{three}−\mathrm{digit}\:\mathrm{number}\:\mathrm{A} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{reverse}\:\mathrm{order}\:\mathrm{to} \\ $$$$\mathrm{form}\:\mathrm{another}\:\mathrm{three}−\mathrm{digit}\:\mathrm{number}\:\mathrm{B}. \\ $$$$\mathrm{If}\:\mathrm{B}>\mathrm{A}\:\mathrm{and}\:\mathrm{B}−\mathrm{A}\:\mathrm{is}\:\mathrm{perfectly} \\ $$$$\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of} \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{A}. \\ $$

Question Number 113002 Answers: 1 Comments: 0

After distributing sweets equally among 25 children, 8 sweets remained. Had the number of children been 28, 22 sweets would have been left after equally distributing. What was the total number of sweets?

$$\mathrm{After}\:\mathrm{distributing}\:\mathrm{sweets}\:\mathrm{equally} \\ $$$$\mathrm{among}\:\mathrm{25}\:\mathrm{children},\:\mathrm{8}\:\mathrm{sweets} \\ $$$$\mathrm{remained}.\:\mathrm{Had}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{children} \\ $$$$\mathrm{been}\:\mathrm{28},\:\mathrm{22}\:\mathrm{sweets}\:\mathrm{would}\:\mathrm{have}\:\mathrm{been} \\ $$$$\mathrm{left}\:\mathrm{after}\:\mathrm{equally}\:\mathrm{distributing}.\:\mathrm{What} \\ $$$$\mathrm{was}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{sweets}? \\ $$

Question Number 113001 Answers: 0 Comments: 1

N! is completely divisible by 13^(52) . What is the sum of the digits of the smallest such number N?

$$\mathrm{N}!\:\mathrm{is}\:\mathrm{completely}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{13}^{\mathrm{52}} . \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{smallest}\:\mathrm{such}\:\mathrm{number}\:\mathrm{N}? \\ $$

Question Number 113000 Answers: 1 Comments: 0

The first 23 natural numbers are written in an increasing order beside each other to form a single number. What is the remainder when this number is divided by 18?

$$\mathrm{The}\:\mathrm{first}\:\mathrm{23}\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{order} \\ $$$$\mathrm{beside}\:\mathrm{each}\:\mathrm{other}\:\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{number}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\mathrm{this}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{18}? \\ $$

Question Number 112999 Answers: 1 Comments: 0

Product of divisors of 7056 ?

$$\mathrm{Product}\:\mathrm{of}\:\mathrm{divisors}\:\mathrm{of}\:\mathrm{7056}\:? \\ $$

Question Number 112998 Answers: 1 Comments: 0

If x+y+z=1 and x,y,z are positive real numbers, then the least value of ((1/x)−1)((1/y)−1)((1/z)−1) is

$$\mathrm{If}\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{1}\:\mathrm{and}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{real}\:\mathrm{numbers},\:\mathrm{then}\:\mathrm{the}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of} \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{x}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{y}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{z}}−\mathrm{1}\right)\:\mathrm{is}\: \\ $$

Question Number 112996 Answers: 1 Comments: 0

Find the last digit of the sum 19^(81) +4^(9k) , k∈N

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{19}^{\mathrm{81}} +\mathrm{4}^{\mathrm{9k}} ,\:\mathrm{k}\in\mathrm{N} \\ $$

Question Number 112994 Answers: 1 Comments: 0

If n is any even number, then n(n^2 +20) is always divisible by?

$$\mathrm{If}\:\mathrm{n}\:\mathrm{is}\:\mathrm{any}\:\mathrm{even}\:\mathrm{number},\:\mathrm{then} \\ $$$$\mathrm{n}\left(\mathrm{n}^{\mathrm{2}} +\mathrm{20}\right)\:\mathrm{is}\:\mathrm{always}\:\mathrm{divisible}\:\mathrm{by}? \\ $$

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