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In this series, we will be updating this post with all the quality questions that we cover on our Facebook and WhatsApp prep groups. Each question is followed with a link to the post that has the answer/solution to it. In case there is some ambiguity, let us know in the comments below or on the relevant post. If you wish to contribute questions so that you have a neat repository for yourself, do let us know. We will be happy to credit you with the question and publish it here. If you haven’t joined the group yet, you may do it here: Cracking CAT with Learningroots

Together, we learn more! So here we go: Numbers Question Bank – CAT 2017

You may go through our other question banks here: CAT 2017 Question Banks

1st edition: 13 September 2017 (Q.1 to Q.54)

# Numbers Question Bank – CAT 2017

1. All the page numbers from a book are added, beginning at page 1. However, one page number was mistakenly added twice. The sum obtained was 1000. Which page number was added twice?

2. What is the product of all the factors of 1800 that have 5 as unit’s digit. Give your answer as a^x*b^y*c^z where a b c are primes

3. M is a two digit number such that the product of the factorials of its individual digits is greater than the sum of the factorials of its individual digits. How many values of M exist?

4. If n is a positive prime number, and n^4 + 3(n^3) is a perfect cube, what is the sum of the lowest two possible values of n?

5. When 6 is affixed at the end of a five-digit number and the resultant number is quadrupled, the result is 6 followed by the original five-digit number. What was the original 5-digit number?

6. Find the greatest number which when divides 378, 2079 and 1890 leaves the same remainder.

7. 120 blue marbles, 180 green marbles and 240 red marbles are to be arranged in rows in such a manner that each row contains an equal number of marbles. Also, each row should have marbles of the same colour. What is the minimum number of rows required?

8. What is the greatest power of 5 which can divide 80! exactly?

9. Simplify 2^73 – 2^72 – 2^71

10. How many 3-digit even numbers can you form such that if one of the digits is 5 then the following digit must be 7?

11. Ria has three types of boxes viz. red, blue and green. She plays a game in which she placed 9 red boxes on the table. She puts 5 blue boxes each, in a few of the red boxes then she puts 5 green boxes each, in few of the blue boxes. If the number of boxes that have been left empty in the game is 41, then how many boxes were used in the game by Ria?

12. Three friends are playing a card game. They start with sums of money in the ratio 7 : 6 : 5 and finish with sums of money in the ratio 6 : 5 : 4, in the same order as before. One of them won Rs. 12. How many rupees did he start with? [The three friends gambled amongst each other only]

13. Atul has exactly six sealed bags containing 15, 31, 19, 20, 16 and 18 candies. Out of the six bags with Atul, there are exactly five bags that contained chocolate candies whereas one box contained orange candies. He distributed all the six bags among his three sons in such a manner that his eldest son got the only box with orange candies and the other bags were distributed in such a manner so that other two brothers received the chocolate candies in the ratio of 2 : 1. How many orange candies were there with Atul? (Assume no candies were taken out of the bags)

14. For how many integers m, is (5m+23)/(m-7) also an integer?

15. A six-digit number is formed by writing 3 consecutive two-digit number side by side in ascending order. If the number so formed is divisible by 2, 3, 4, 5, 6, 8, then what is the hundreds digit of the number?

16. The year 1789 has three and no more than three adjacent digits (7, 8 and 9) which are consecutive integers in increasing order. How many years between 1000 and 9999 have this property?

17. A survey of 104 students, who study Physics or Mathematics or Chemistry, revealed that 38 did not study Physics, 30 did not study Mathematics and 40 did not study Chemistry. Also 40 studied Chemistry and Physics, 50 studied Chemistry and Mathematics and 44 studied Physics and Mathematics. What is the difference between the number of students who studied Mathematics and the number of students who studied exactly two of the three subjects?

18. At the FMS Alumni meet of 2017, each alumnus brought either 1 or 2 family members. If the ratio of the number of alumni to the number of family members is 3 : 5, what fraction of the alumni brought 2 guests?

19. Diophantus’s youth lasted one sixth of his life. He grew a beard after one twelfth more. After one seventh more of his life, he married. 5 years later, he and his wife had a son. The son lived exactly one half as long as his father, and Diophantus died four years after his son. How many years did Diophantus live?

20. Mr. X went to the C-Mart near his house and purchased four chocolates from there. The first one cost Rs. 1.5, the second one cost Rs. 3, and the third one cost Rs. 4. When Mr. X multiplied the prices of the four chocolates he got a product which was the same as the bill amount in Rupees he paid at C-Mart. We know that he paid a Rs. 50 note at the counter and got back some money. If the billing executive gave him the minimum number of notes/coins, then Mr. X has surely got which of the following notes/coins back from the billing executive?

21. The sum of a two-digit number and the number formed by interchanging the two digits is 45 more than twice the original number. If the sum of the digits of the number is 9, what is the original number?

22. How many three digit numbers are of the form xyz with x < y, z < y and x ≠ 0?

23. There is a staircase of 10 steps. In how many ways can Amit climb the staircase if he can take a maximum of 3 steps at a time?

24. What is the sum of factors of each factor of 1024?

25. Find the number of ordered pairs (x, y) where both x and y are non-negative integers such that,

x – (1/y) = (x/y) + 1?

26. A boy is asked to take a 2-digit number; multiply its digits and keep repeating the process until he gets a single digit number. What is the probability that this single digit is zero?

27. In how many ways can 720 be expressed as product of 2 co-primes?

28. How many pairs of factors of 720 exist such that the factors are co-prime to each other?

29. A set N is formed by selecting some of the numbers from the first 110 natural numbers such that the GCD of any two numbers in the set is 5. What is the maximum number of elements that set N can have?

30. The number of 6-digit numbers of the form ababab (where a and b are distinct, non-negative integers) each of which is a product of exactly 6 district primes is

31. The population of cattle in a farm increases so that the difference between the population in year n + 2 and that in year n is proportional to the population in year n + 1. If the populations in year 2014, 2015 and 2017 were 39, 60 and 123, respectively, then the population in 2016 was

32. MTNL has a waiting list of 5005 applicants for its recently launched mobile phone scheme. The list shows that there are at least 5 males between any two females. The largest possible number of females in the waiting list is:

33. During a parade 289 soldiers are standing in a square formation with 17 ranks and 17 files. Bullets were handed out to each of these soldiers for target practice. The number of bullets with the soldiers in each rank, as well as in each file, were in an arithmetic progression. If the number of bullets with the 4th and the 14th soldiers in the first rank were 831 and 861 respectively, while the number of bullets with the 2nd and the 16th soldiers in the 16th rank were 60 and 102 respectively, find the average number of bullets with all the soldiers.

34. Given that 2^x^y + x^2^y = 3, how many integral solutions are there for the equation?

35. There are 5 distinct real numbers out of which all possible triplets are selected and for each triplet the three numbers are added. The different sums that are generated are: (– 8, 1, 3, 5, 7, 8, 10, 16, 19 and 23).

The smallest among the 5 numbers is

36. There are 5 distinct real numbers out of which all possible triplets are selected and for each triplet the three numbers are added. The different sums that are generated are: (– 8, 1, 3, 5, 7, 8, 10, 16, 19 and 23).

The third largest number is

37. There are 98 given points on a circle. Amar, Akbar and Anthony start playing a game by drawing a chord one by one between two of the points which have not yet been joined together. The game ends when all such points have been joined exhaustively. The winner is the one who draws the last chord. If Anthony starts the game, followed by Akbar, and then Amar, then who will win?

38. A, B, C, D, E and F are six single-digit non-negative integers such that A < B < C < D < E < F. Three-digit number CFC is a perfect square, BE is a two-digit prime number and A + D + F = B + C + E.

What is the value of D?

a. 4
b. 5
c. 6
d. Cannot be determined

39. A, B, C, D, E and F are six single-digit non-negative integers such that A < B < C < D < E < F. Three-digit number CFC is a perfect square, BE is a two-digit prime number and A + D + F = B + C + E.

The four-digit natural number BEFC is definitely not divisible by which of the following two-digit numbers?

a. CB
b. AA
c. CC
d. CE

40. An instruction was sent to all the railway stations across the country that, workers above the age of 47 years must retire. Five workers with different ages at the Kazipet railway station managed to tamper their records. However, the station master knew that the sums of the ages (in years) of all the possible pairs of workers (from out of the five) are 102, 105, 107, 107, 109, 109, 111, 112, 114, 116. Find the age of the oldest of the five workers.

41. X toffees can be distributed equally among Y children, where X > Y. What is the number of Values that X can assume such that Y > 1 and 2 < X + Y < 110?

42. If x^2 < 81 and y^2 < 25, what is the largest prime number that can be equal to x – 2y?

43. A password on Mr. Marsellus Wallace’s briefcase consists of 5 digits. What is the probability that the password contains exactly three sixes?

44. x, y and z are positive integers such that when x is divided by y the remainder is 3 and when y is divided by z the remainder is 8. What is the smallest possible value of x+y+z?

45. What is the smallest positive integer k such that 126*sqrt(k) is the square of a positive integer?

46. Of the applicants passes a certain test, 15 applied to both college X and Y. If 20 % of the applicants who applied college X and 25% of the applicants who applied college Y applied both college X and Y, how many applicants applied only college X or college Y?

47. The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is prime, then f(p) = ?

(A) P-1
(B) P-2
(C) (P+1)/2
(D) (P-1)/2

48. N is an eight digit number and S(N) is the sum of the digits of N. If N + S(N) = 100,000,000 what would be N?

49. N represents a series in which all the terms are consecutive integers and the sum of all the terms of N is 100. If the number of terms of N is greater than one, find the difference between the maximum and the minimum possible number of terms.

50. If a1 = 2, a2 = 3 and an+2 + an = 2an+1 + 1 for every positive integer n, then a51 equals?

51. If [log1] + [log2] + [log3] + [log4] … + [logn] = n where [x] denotes the greatest integer less than or equal to x, then which of the following is an acceptable range for n? (all logs are in base 10)

a. 96 < n < 104

b. 104 < n < 107

c. 107 < n < 111

d. 111 < n < 116

52. M/30! = 1/30! + 1/29! + 1/2!28! + 1/3!27! + … + 1/28!2! + 1/29! + 1/30!. Find the quotient when M – 1 is divided by 1023.

53. If n is a natural number and n! = n(n – 1)(n – 2)…3.2.1, find the remainder when summation of n(n!) is divided by n2 – 2n (n > 2)

a. 0

b. n

c. n2 – 2n – 1

d. (n – 1)2

54. The sum of all positive integers n for which (13 + 23 + 33 … (2n)3)/(12 + 22 + 32 … n2) is also an integer is?