Question Bank - Number Theory - Shashank Prabhu, CAT 100 Percentiler
Q46) Let x, y, z be three non-negative integers such that x + y + z = 10. The maximum possible value of xyz + xy + yz + zx is
Q47) Marx, Max, Meryll and Minerva won a certain number of points in a shooting competition. Minerva secured half the number of points won by other three. Meryll won one-third the numbers of points accumulated by the other three, while Marx got two-fifth the number of points won by the other three. If each of them won a distinct number of points, find the ratio of the number of points secured by Max to the total number of points won by all four.
d. Cannot be determined uniquely
[OA: Option a]
Q48) Number 22804 is written in a base 10 is written as a three digit number mmm in base b where 1 ≤ m ≤ 9. What is the value of b ?
22804=4*5701.. factors are 1, 2, 4, 5701, 11402, 22804
b^2+b+1 cannot be 1, 2 or 4.
b(b+1) has to be either 5700, 11401 or 22803.. product of 2 consecutive numbers will always be an even number. So, m=4 and b(b+1)=5700.. b=75
Q49) A palindrome is a number which reads same forward and backward, e.g. 121 is a three digit palindrome number. What is the sum of all three digit palindromes which are multiples of 13?
100a + 10b + a = 13k
101a + 10b = 13k
91a + 10(a + b) = 13k
10(a+b) has to be a multiple of 13. a+b has to be a multiple of 13. Find out the numbers and add.
Q50) 9 positive numbers are arranged around a circle. Each number is one more than the greatest common divisors of its two neighbours. The minimum sum of these numbers is
Q51) Consider the numbers 3, 8, 13… 103, 108. What is the smallest value of n such that every collection of n of these numbers will always contain a pair which sums to 121?
Q53) Find the sum of all prime numbers p for which 5p + 1 is a perfect square.
Q57) What is the smallest possible difference between a square number and a prime number, if prime is greater than 3 and the square number is greater than prime?
Q65) Avnish counted the digits used to number the pages of a book and found that the total number of digits used was 3441. Find the number of pages in the book.
Q67) How many natural numbers are factors of at least one of 6^14, 10^12 and 15^20?
(45-9) = ksqrt(16), k=9
45 = 9sqrt(n)
n = 25
At 25, it will stop. So, 24 can be carried
Q71) Lara is deciding whether to visit Kullu or Cherapunji for the holidays. She makes her decision by rolling a regular 6-sided die. If she gets a 1 or 2, she goes to Kullu. If she rolls a 3, 4, or 5, she goes to Cherapunji. If she rolls a 6, she rolls again. What is the probability that she goes to Cherapunji?
Q72) How many 9-digit numbers (in decimal system) divisible by 11 are there in which every digit occurs except zero? [Credits : TG]
Q73) In a box containing 15 apples, exactly 6 apples are rotten. Each day one apple is taken out from the box. What is the probability that after four days there are exactly 8 apples in the box that are not rotten?
d. None of these
[OA: Option a]
Q74) What is the remainder when n! + (n! + 1) + (n! – 2) + (n! + 3) ..... + (n! – 2006) is divided by 1003 for n = 1003?
Q75) What is the 10th positive integer that cannot be expressed as the sum of two or more consecutive positive integers?
Q76) What is the 2037th positive integer that can be expressed as the sum of two or more consecutive positive integers?
Q77) No. of positive integral solutions for 3/x - 7/y = 1/14