Number Theory Previous Year Questions (CAT)  Set 0011

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

Question 1
Let D be a recurring decimal of the form, D = 0.a_{1}a_{2}a_{1}a_{2}a_{1}a_{2 }......., where digits a_{1 }and a_{2 }lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?
(1) 18
(2) 108
(3) 198
(4) 288 (CAT 2000)

Answer: Option 3

Question 2
If a_{1}= 1 and a_{n}_{+1 }= 2a_{n}+ 5, n = 1, 2 ... , then a_{100} is equal to
(1) (5 × 2^{99} – 6)
(2) (5 × 2^{99} + 6)
(3) (6 × 2^{99} + 5)
(4) (6 × 2^{99} – 5) (CAT 2000)

Question 3
(CAT 2000)
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Question 4
If x > 2 and y > – 1, Then which of the following statements is necessarily true?
(1) xy > –2
(2) –x < 2y
(3) xy < –2
(4) –x > 2y (CAT 2000)

Answer: Option 2

Question 5
(CAT 2000)

Question 6
Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all elements of S. With how many consecutive zeros will the product end?
(1) 1
(2) 4
(3) 5
(4) 10 (CAT 2000)

Question 7
Let N = 1421 × 1423 × 1425. What is the remainder when N is divided by 12?
(1) 0
(2) 9
(3) 3
(4) 6 (CAT 2000)

Answer: Option 3

Question 8
Each of the numbers x_{1}, x_{2}...., x_{n}, n > 4, is equal to 1 or –1.
Suppose, x_{1}x_{2}x_{3}x_{4 }+ x_{2}x_{3}x_{4}x_{5 }+ x_{3}x_{4}x_{5}x_{6 }+ ... + x_{n–3}x_{n–2}x_{n–1}x_{n} + x_{n–2}x_{n–1}x_{n}x_{1}+ x_{n–1}x_{n}x_{1}x_{2 }+ x_{n}x_{1}x_{2}x_{3}= 0, then,(1) n is even.
(2) n is odd.
(3) n is an odd multiple of 3.
(4) n is prime (CAT 2000)

Question 9
The integers 34041 and 32506 when divided by a threedigit integer n leave the same remainder. What is n?
(1) 289
(2) 367
(3) 453
(4) 307 (CAT 2000)