Number Theory Previous Year Questions (CAT)  Set 0005

Previous years`s CAT questions: Number Theory
Post your solutions as reply to respective questions below.

Question 1
Let n! = 1 × 2 × 3 × ... × n for integer n ≥ 1. If p = 1! + (2 × 2!) + (3 × 3!) + … + (10 × 10! ), then p + 2 when divided by 11! leaves a remainder of
(1) 10
(2) 0
(3) 7
(4) 1 (CAT 2005)

Answer: Option 4

Question 2
The digits of a three digit number A are written in the reverse order to form another three digit number B. If B > A and B − A is perfectly divisible by 7, then which of the following is necessarily true?
(1) 100 < A < 299
(2) 106 < A < 305
(3) 112 < A < 311
(4) 118 < A < 317 (CAT 2005)

Answer: Option 2

Question 3
If a_{1} = 1 and a_{n }_{+ 1} – 3a_{n} + 2 = 4n for every positive integer n, then a_{100} equals
(1) 3^{99} – 200
(2) 3^{99} + 200
(3) 3^{100} – 200
(4) 3^{100} + 200 (CAT 2005)

Answer: Option 3

Question 4
The rightmost non zero digit of the number 30^{2720} is
(1) 1
(2) 3
(3) 7
(4) 9 (CAT 2005)

Answer: Option 1

Question 5
For a positive integer n, let P_{n} denote the product of the digits of n, and S_{n} denote the sum of the digits of n. The number of integers between 10 and 1000 for which P_{n} + S_{n} = n is
(1) 81
(2) 16
(3) 18
(4) 9 (CAT 2005)

Answer: Option 4

Question 6
Let S be a set of positive integers such that every element n of S satisfies the conditions
a) 1000 ≤ n ≤ 1200
b) every digit in n is odd.
Then how many elements of S are divisible by 3?
(1) 9
(2) 10
(3) 11
(4) 12 (CAT 2005)

Answer: Option 1

Question 7
The 288^{th }term of the series a, b, b, c, c, c, d, d, d, d,e, e, e, e, e, f, f, f, f, f, f... is
(1) u
(2) v
(3) w
(4) x (CAT 2003 Leaked)

Answer: Option 4

Question 8
A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs. 250 and Rs. 300 per day respectively. In addition, a male operator gets Rs. 15 per call he answers and a female operator gets Rs. 10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?
(1) 15
(2) 14
(3) 12
(4) 10 (CAT 2005)

Answer: Option 4

Question 9
Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over persontoperson phone calls so that eventually each person knows all six secrets. None ?of the Frenchmen knows English, and only one Englishman knows French. What is the minimum number of phone calls needed for the above purpose?
(1) 5
(2) 10
(3) 9
(4) 15 (CAT 2005)

Answer: Option 3

Question 10
The total number of integer pairs (x, y) satisfying the equation x + y = xy is
1. 0
2. 1
3. 2
4. None of the above (CAT 2004)