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Number TheoryQuestion and Answers: Page 20
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Question and Answers Forum |
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Number TheoryQuestion and Answers: Page 20 |
| 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? |
| 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? |
| 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. |
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| if :∀ε>0, ∀(a,b)∈R^2 ,a<b+ε prove: a≤b |
| 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. |
| What is the digital root of 3^(2017) |
| 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. |
| The sum of two positive integers is 52 and their LCM is 168. Find the numbers. |
| Find a natural number ′n′ such that 3^9 + 3^(12) + 3^(15) + 3^n is a perfect cube of an integer. |
| what is the maximum number of time three divides 333^(505) |
| Find the number of numbers ≤ 10^8 which are neither perfect squares, nor perfect cubes, nor perfect fifth powers. |
| 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. |
| Find the number of odd integers between 30,000 and 80,000 in which no digit is repeated. |
| Show that for any natural number n, the fraction ((21n + 4)/(14n + 3)) is in its lowest term. |
| How many times is digit 0 written when listing all numbers from 1 to 3333? |
| Let x be the LCM of 3^(2002) − 1 and 3^(2002) + 1. Find the last digit of x. |
| Find the integer closest to 100(12 − (√(143))). |
| What are next three numbers in the following sequence: 4,6,12,18,30,42,60,... |
| Determine two distinct primes p and q such that: (i) p+q+1,p+q−1,((p+q)/2) ∈ P (All primes)? (ii) p+q+1,p+q−1,((p+q)/2),((p−q)/2) ∈ P (All primes)? |
| The sum of the digits of the number 2^(2000) 5^(2004) is Will it be 13 or 14? |
| Find two primes a and b such that a−b=995 |
| Find the number of positive integers less than or equal to 300 that are multiples of 3 or 5, but are not multiples of 10 or 15. |
| Find out last odd digit in the expansion of 1000! |
| Find out first non-five digit from right in the expansion of (1×3×5×...×625). |
| Number of decimal digits in 50! is |