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Number TheoryQuestion and Answers: Page 20

Question Number 21781 Answers: 2 Comments: 0

Which is greater 10^(11) or 11^(10) ?

$$\mathrm{Which}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{10}^{\mathrm{11}} \:\mathrm{or}\:\mathrm{11}^{\mathrm{10}} ? \\ $$

Question Number 21682 Answers: 1 Comments: 0

Prove that the ten′s digit of any power of 3 is even. [e.g. the ten′s digit of 3^6 = 729 is 2].

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{ten}'\mathrm{s}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{any}\:\mathrm{power} \\ $$$$\mathrm{of}\:\mathrm{3}\:\mathrm{is}\:\mathrm{even}.\:\left[\mathrm{e}.\mathrm{g}.\:\mathrm{the}\:\mathrm{ten}'\mathrm{s}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{3}^{\mathrm{6}} \:=\right. \\ $$$$\left.\mathrm{729}\:\mathrm{is}\:\mathrm{2}\right]. \\ $$

Question Number 21573 Answers: 0 Comments: 0

Prove that 1 < (1/(1001)) + (1/(1002)) + (1/(1003)) + ... + (1/(3001)) < 1(1/3).

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{1}\:<\:\frac{\mathrm{1}}{\mathrm{1001}}\:+\:\frac{\mathrm{1}}{\mathrm{1002}}\:+\:\frac{\mathrm{1}}{\mathrm{1003}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{3001}}\:<\:\mathrm{1}\frac{\mathrm{1}}{\mathrm{3}}. \\ $$

Question Number 21423 Answers: 1 Comments: 0

Prove that n^4 + 4^n is composite for all integer values of n greater than 1.

$$\mathrm{Prove}\:\mathrm{that}\:{n}^{\mathrm{4}} \:+\:\mathrm{4}^{{n}} \:\mathrm{is}\:\mathrm{composite}\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:{n}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{1}. \\ $$

Question Number 21292 Answers: 0 Comments: 0

Let a,b∈Z 0<a<b How would you find the maximum/ largest prime gap in (a, b)? Note: Prime gaps are the distance between consecutive primes. e.g. 7 and 11 has a prime gap 4 p_k ∈P ∴∀p_x ∀p_(x+1) ∈(a,b):p_(x+1) >p_x p_(x+1) and p_x are consecutive primes Lets denote δ_x =p_(x+1) −p_x as prime gap for (1, 20), the primes are 2,3,5,7,11,13,17 The prime gaps are: 1,2,2,4,2,4 Therefore the largest δ = 4 Is there a more general method?

$$\mathrm{Let}\:\:{a},{b}\in\mathbb{Z} \\ $$$$\mathrm{0}<{a}<{b} \\ $$$$\: \\ $$$$\mathrm{How}\:\mathrm{would}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum}/ \\ $$$$\mathrm{largest}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{in}\:\left({a},\:{b}\right)? \\ $$$$ \\ $$$$\mathrm{Note}: \\ $$$$\mathrm{Prime}\:\mathrm{gaps}\:\mathrm{are}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{consecutive}\:\mathrm{primes}. \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\:\mathrm{7}\:\mathrm{and}\:\mathrm{11}\:\mathrm{has}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{4} \\ $$$$\: \\ $$$${p}_{{k}} \in\mathbb{P} \\ $$$$\therefore\forall{p}_{{x}} \forall{p}_{{x}+\mathrm{1}} \in\left({a},{b}\right):{p}_{{x}+\mathrm{1}} >{p}_{{x}} \\ $$$${p}_{{x}+\mathrm{1}} \:\mathrm{and}\:{p}_{{x}} \:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{primes} \\ $$$$\mathrm{Lets}\:\mathrm{denote}\:\delta_{{x}} ={p}_{{x}+\mathrm{1}} −{p}_{{x}} \:\mathrm{as}\:\mathrm{prime}\:\mathrm{gap} \\ $$$$\: \\ $$$$\mathrm{for}\:\left(\mathrm{1},\:\mathrm{20}\right),\:\mathrm{the}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17} \\ $$$$\mathrm{The}\:\mathrm{prime}\:\mathrm{gaps}\:\mathrm{are}: \\ $$$$\mathrm{1},\mathrm{2},\mathrm{2},\mathrm{4},\mathrm{2},\mathrm{4} \\ $$$$\mathrm{Therefore}\:\mathrm{the}\:\mathrm{largest}\:\delta\:=\:\mathrm{4} \\ $$$$\: \\ $$$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{more}\:\mathrm{general}\:\mathrm{method}? \\ $$

Question Number 21353 Answers: 1 Comments: 0

A censusman on duty visited a house which the lady inmates declined to reveal their individual ages, but said − “we do not mind giving you the sum of the ages of any two ladies you may choose”. Thereupon the censusman said − “In that case please give me the sum of the ages of every possible pair of you”. The gave the sums as follows : 30, 33, 41, 58, 66, 69. The censusman took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?

Question Number 21229 Answers: 0 Comments: 0

Let p, q be prime numbers such that n^(3pq) − n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.

$$\mathrm{Let}\:{p},\:{q}\:\mathrm{be}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$${n}^{\mathrm{3}{pq}} \:−\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3}{pq}\:\mathrm{for}\:\boldsymbol{\mathrm{all}} \\ $$$$\mathrm{positive}\:\mathrm{integers}\:{n}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{p}\:+\:{q}. \\ $$

Question Number 21053 Answers: 0 Comments: 0

Question Number 21031 Answers: 0 Comments: 0

if :∀ε>0, ∀(a,b)∈R^2 ,a<b+ε prove: a≤b

$${if}\::\forall\epsilon>\mathrm{0},\:\forall\left({a},{b}\right)\in\mathbb{R}^{\mathrm{2}} ,{a}<{b}+\epsilon \\ $$$${prove}:\:{a}\leqslant{b} \\ $$

Question Number 19631 Answers: 1 Comments: 0

Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product then a number x is obtained which is a multiple of 17. Find the sum of digits of number x.

$$\mathrm{Two}\:\mathrm{different}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{between} \\ $$$$\mathrm{4}\:\mathrm{and}\:\mathrm{18}\:\mathrm{are}\:\mathrm{chosen}.\:\mathrm{When}\:\mathrm{their}\:\mathrm{sum}\:\mathrm{is} \\ $$$$\mathrm{subtracted}\:\mathrm{from}\:\mathrm{their}\:\mathrm{product}\:\mathrm{then}\:\mathrm{a} \\ $$$$\mathrm{number}\:{x}\:\mathrm{is}\:\mathrm{obtained}\:\mathrm{which}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{multiple}\:\mathrm{of}\:\mathrm{17}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{of} \\ $$$$\mathrm{number}\:{x}. \\ $$

Question Number 19389 Answers: 1 Comments: 0

What is the digital root of 3^(2017)

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{digital}\:\mathrm{root}\:\mathrm{of}\:\:\mathrm{3}^{\mathrm{2017}} \\ $$

Question Number 19239 Answers: 1 Comments: 0

Assume that a, b, c and d are positive integers such that a^5 = b^4 , c^3 = d^2 and c − a = 19. Determine d − b.

$$\mathrm{Assume}\:\mathrm{that}\:{a},\:{b},\:{c}\:\mathrm{and}\:{d}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:{a}^{\mathrm{5}} \:=\:{b}^{\mathrm{4}} ,\:{c}^{\mathrm{3}} \:=\:{d}^{\mathrm{2}} \:\mathrm{and} \\ $$$${c}\:−\:{a}\:=\:\mathrm{19}.\:\mathrm{Determine}\:{d}\:−\:{b}. \\ $$

Question Number 19193 Answers: 1 Comments: 0

The sum of two positive integers is 52 and their LCM is 168. Find the numbers.

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{is}\:\mathrm{52} \\ $$$$\mathrm{and}\:\mathrm{their}\:\mathrm{LCM}\:\mathrm{is}\:\mathrm{168}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{numbers}. \\ $$

Question Number 19192 Answers: 0 Comments: 2

Find a natural number ′n′ such that 3^9 + 3^(12) + 3^(15) + 3^n is a perfect cube of an integer.

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:'\mathrm{n}'\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{3}^{\mathrm{9}} \:+\:\mathrm{3}^{\mathrm{12}} \:+\:\mathrm{3}^{\mathrm{15}} \:+\:\mathrm{3}^{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{cube}\:\mathrm{of} \\ $$$$\mathrm{an}\:\mathrm{integer}. \\ $$

Question Number 19002 Answers: 1 Comments: 0

what is the maximum number of time three divides 333^(505)

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{number}\:\mathrm{of}\:\:\mathrm{time}\:\mathrm{three}\:\mathrm{divides}\:\mathrm{333}^{\mathrm{505}} \\ $$

Question Number 18949 Answers: 1 Comments: 1

Find the number of numbers ≤ 10^8 which are neither perfect squares, nor perfect cubes, nor perfect fifth powers.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{numbers}\:\leqslant\:\mathrm{10}^{\mathrm{8}} \\ $$$$\mathrm{which}\:\mathrm{are}\:\mathrm{neither}\:\mathrm{perfect}\:\mathrm{squares},\:\mathrm{nor} \\ $$$$\mathrm{perfect}\:\mathrm{cubes},\:\mathrm{nor}\:\mathrm{perfect}\:\mathrm{fifth}\:\mathrm{powers}. \\ $$

Question Number 18884 Answers: 1 Comments: 0

Determine the smallest positive integer x, whose last digit is 6 and if we erase this 6 and put it in left most of the number so obtained, the number becomes 4x.

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer} \\ $$$$\mathrm{x},\:\mathrm{whose}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{6}\:\mathrm{and}\:\mathrm{if}\:\mathrm{we}\:\mathrm{erase} \\ $$$$\mathrm{this}\:\mathrm{6}\:\mathrm{and}\:\mathrm{put}\:\mathrm{it}\:\mathrm{in}\:\mathrm{left}\:\mathrm{most}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{so}\:\mathrm{obtained},\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{becomes}\:\mathrm{4x}. \\ $$

Question Number 18655 Answers: 0 Comments: 3

Find the number of odd integers between 30,000 and 80,000 in which no digit is repeated.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{integers} \\ $$$$\mathrm{between}\:\mathrm{30},\mathrm{000}\:\mathrm{and}\:\mathrm{80},\mathrm{000}\:\mathrm{in}\:\mathrm{which}\:\mathrm{no} \\ $$$$\mathrm{digit}\:\mathrm{is}\:\mathrm{repeated}. \\ $$

Question Number 18652 Answers: 1 Comments: 0

Show that for any natural number n, the fraction ((21n + 4)/(14n + 3)) is in its lowest term.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{natural}\:\mathrm{number}\:{n}, \\ $$$$\mathrm{the}\:\mathrm{fraction}\:\frac{\mathrm{21}{n}\:+\:\mathrm{4}}{\mathrm{14}{n}\:+\:\mathrm{3}}\:\mathrm{is}\:\mathrm{in}\:\mathrm{its}\:\mathrm{lowest}\:\mathrm{term}. \\ $$

Question Number 18498 Answers: 5 Comments: 1

How many times is digit 0 written when listing all numbers from 1 to 3333?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{is}\:\mathrm{digit}\:\mathrm{0}\:\mathrm{written}\:\mathrm{when} \\ $$$$\mathrm{listing}\:\mathrm{all}\:\mathrm{numbers}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{3333}? \\ $$

Question Number 18135 Answers: 0 Comments: 6

Let x be the LCM of 3^(2002) − 1 and 3^(2002) + 1. Find the last digit of x.