# 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

[OA: 69]

• 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.
a. 11:84
b. 17:42
c. 11:73
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 ?

[OA: 75]

• m(b^2+b+1)=22804
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?

[OA: 4329]

• 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

[OA: 21]

• 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?

[OA: 13]

• Q53) Find the sum of all prime numbers p for which 5p + 1 is a perfect square.

[OA: 10]

• 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?

[OA: 2]

• 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.

[OA: 1137]

• Q67) How many natural numbers are factors of at least one of 6^14, 10^12 and 15^20?

[OA: 795]

• (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?

[OA: 3/5]

• Q72) How many 9-digit numbers (in decimal system) divisible by 11 are there in which every digit occurs except zero? [Credits : TG]

[OA: 31680]

• 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?
a. 12/91
b. 1/7
c. 2/13
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?

[OA: 0]

• Q75) What is the 10th positive integer that cannot be expressed as the sum of two or more consecutive positive integers?

[OA: 512]

• Q76) What is the 2037th positive integer that can be expressed as the sum of two or more consecutive positive integers?

[OA: 2049]

• Q77) No. of positive integral solutions for 3/x - 7/y = 1/14

[OA: 10]

102

86

45

58

65

68

45