Question Bank  Number theory  Sibanand Pattnaik

Q63) The smallest number of the form 6k (k is a natural number), such that both 6k + 1 and 6k  1 are composite is?

Q64) A box contains 100 tickets, numbered from 1 to 100. A person picks out three tickets from the box, such that the product of the numbers on two of the tickets yields the number on the third ticket. How many tickets can never be picked up?
a. 26
b. 10
c. 11
d. 25

Q65) Set A is formed by selecting some of the numbers from the first 100 natural numbers such that the HCF of any two numbers in the set is the same.
If every pair of numbers for set A has to be relatively prime and set A has the maximum numbers of
elements possible, then in how ways can the set A be selected?
a. 64
b. 96
c. 72
d. 108If the HCF of any two numbers in set A is 3, what is the maximum number elements that set A can
have?
a. 10
b. 12
c. 11
d. 14

Q66) The sum of squares of three consecutive odd natural numbers is bbbb, a fourdigit number. Find the
value of b.
a) 1
b) 2
c) 5
d) 8

Q67) How many positive even integers less than 200 can be written as the sum of three consecutive
integers?
a) 33
b) 20
c) 15
d) 35

Q68) Find the number of twodigit natural numbers with the following characteristics:
I. Number should be even.
II. Number should have exactly four factors including 1 and the number itself.
III. One of the digits of the number should be a perfect square.
IV. If we reverse the digits of the number, we should receive a prime number.
V. The sum of digits of the number should be a prime.
a) 0
b) 1
c) 2
d) 3

Q69) Raju went to a shop to buy a certain number of pens and pencils. He calculated the amount payable
to the shopkeeper and gave that amount to him. Raju was surprised when the shopkeeper returned Rs. 24,
to him, as balance. When he came back home, he realized that the shopkeeper had actually transposed the
number of pens with the number of pencils. Which of the following is certainly an invalid statement?
a) The number of pencils that Raju wanted to buy was 8 more than the number of pens.
b) The number of pens that Raju wanted to buy was 6 less than the number of pencils.
c) A pen cost Rs.4 more than a pencil.
d) None of these

Q70) How many 4digit positive integral numbers are there in base 7, if the number of such numbers is
converted to the same base?
a) 2058
b) 5666
c) 6000
d) None of these

Q71) All the positive integers in which sum of digits is equal to the number of digits in the number are
said to be 'pure numbers'. For example, 1, 11 and 20 are pure numbers. How many pure numbers have
more than one digit and all the digits are distinct?
a) 2
b) 3
c) 5
d) Infinite

Q72) Let n = 999...99 be an integer consisting of a string of 2009 nines. Find the sum of digits of n^2
a) 18072
b) 18081
c) 18090
d) 18080
e) 18073

Q73) A set S consists of 143 natural numbers, each of which is a perfect cube. The maximum number of
elements of S that one can always find such that each of them leaves the same remainder when divided by 13 is
a. 29
b. 28
c. 26
d. 27

Q74) HCF of how many distinct pairs of factors of 18000 is 75 ?

Q75) How many factors (x,y,z) of 1000 exist such that HCF (x, y, z) = 1

Q76) Find the number of divisors of 10800 that are divisible by 12 but not divisible by 36

Q77) The sum of 6 natural numbers (not necessarily distinct) is 23. Let M denote the positive difference of the maximum and minimum of the LCM of these 6 numbers. Then total number of divisors that divide M^2 but doesn't divide M are
(a) 22
(b) 33
(c) 18
(d) 26
(e) none of the foregoing

Q78) How many integers satisfy the condition 3^21 < n^14 < 2^63

Q79) How many ordered triples are there if x + y + z = 7

Q80) If for N values of p, where p ≤ 250, the highest power of p in p! is 5, which of the following is true?
a) N ≤ 4
b) N = 6
c) N = 8
d) N ≥ 12

Q81) When a three digit number is divided by the sum of the digits of the number, the quotient is 26. What is the least number for which this is true?

Q82) Sarah intended to multiply a twodigit number and a threedigit number, but she left out the multiplication sign and simply placed the twodigit number to the left of the threedigit number, thereby forming a vedigit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the twodigit number and the threedigit number?