Number Theory Previous Year Questions (CAT)  Set 0008

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

Question 1
If three positive real numbers x, y, z satisfy y – x = z – y and x y z = 4, then what is the minimum possible value of y?
1. 2^{1/3 }
2. 2^{2/3}
3. 2^{1/4}
4.2^{3/4 }(CAT 2003)

Answer: Option 2

Question 2
An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, …, 9 such that the first digit of the code is nonzero. The code, handwritten on a slip, can however potentially create confusion when read upside down — for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?
1. 80
2. 78
3. 71
4. 69 (CAT 2003)

Answer: Option 3

Question 3
Let x and y be positive integers such that x is prime and y is composite. Then,
1. y – x cannot be an even integer.
2. xy cannot be an even integer.
3. (x + y)/x cannot be an even integer.
4. None of the above statements is true. (CAT 2003)

Answer: Option 4

Question 4
If 13x + 1 < 2z, and z + 3 = 5y^{2}, then:
1. x is necessarily less than y.
2. x is necessarily greater than y.
3. x is necessarily equal to y.
4. None of the above is necessarily true.Â Â Â (CAT 2003)

Answer: Option 4

Question 5

Answer: Option 4

Question 6
If b > 1 and x = – a b, then which one of the following is necessarily true?
1. a – xb < 0
2. a – xb ≥ 0
3. a – xb > 0
4. a – xb ≤ 0 (CAT 2003)

Answer: Option 2

Question 7
Let a, b, c, d, and e be integers such that a = 6b = 12c, and 2b = 9d = 12e Then which of the following pairs contains a number that is not an integer?
(1) (a/27, b/e)
(2) (a/36, c/e)
(3) (a/12, bd/18 )
(4) (a/6, c/d) (CAT 2003)

Answer: Option 4

Question 8
If a, a + 2, and a + 4 are prime numbers, then the number of possible solutions for a is:
1. one
2. two
3. three
4. more than three (CAT 2003)

Answer: Option 1

Question 9
On dividing a number by 3, 4 and 7, the remainders are 2, 1 and 4 respectively. If the same number is divided by 84 then the remainder is
(1) 80
(2) 76
(3) 53
(4) None of these (CAT 2002)

Answer: Option 3

Question 10
There are three pieces of cake weighing 9/2 lbs, 27/4 lbs and 36/5 lbs. Pieces of the cake are equally divided and distributed in such a manner that every guest in the party gets one single piece of cake. Further the weight of the pieces of the cake is as heavy as possible. What is the largest number of guest to whom we can distribute the cake?
(1) 54
(2) 72
(3) 20
(4) None of these (CAT 2002)