Number Theory Previous Year Questions (CAT)  Set 0018

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

Question 1
Suppose one wishes to find distinct positive integers x, y such that (x + y)/ xy is also a positive integer. Identify the correct alternative.
(a) This is never possible.
(b) This is possible and the pair (x,y) satisfying the stated condition is unique.
(c) This is possible and there exist more than one but a finite number of ways of choosing the pair (x,y).
(d) This is possible and the pair (x,y) can be chosen in infinite ways.Â (CAT 1993)

Question 2

Question 3
139 persons have signed up for an elimination tournament. All players are to be paired up for the first round, but because 139 is an odd number one player gets a bye, which promotes him to the second round, without actually playing in the first round. The pairing continues on the next round, with a bye to any player left over. If the schedule is planned so that a minimum number of matches is required to determine the champion, the number of matches which must be played is
(a) 136
(b) 137
(c) 138
(d) 139 (CAT 1993)

Question 4
There are ten 50 paise coins placed on a table. Six of these show tails four show heads. A coin is chosen at random and flipped over (not tossed). This operation is performed seven times. One of the coins is then covered. Of the remaining nine coins, five show tails and four show heads. The covered coin shows
(a) a head
(b) a tail
(c) more likely a head
(d) more likely a tail (CAT 1993)

Question 5
The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is
(a) 26
(b) 18
(c) 31
(d) None Â Â (CAT 1993)

Question 6

Question 7
Let x < 0.50, 0 < y < 1, z > 1. Given a set of numbers, the middle number, when they are arranged in ascending order, is called the median. So the median of the numbers x, y, and z would be
(a) less than one
(b) between 0 and 1
(c) greater than 1
(d) cannot sayÂ (CAT 1993)

Question 8
A young girl counted in the following way on the fingers of her left hand. She started calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, index finger 8, thumb 9, then back to the index finger for 10, middle finger for 11, and so on. She counted up to 1994. She ended on her.
(a) thumb
(b) index finger
(c) middle finger
(d) ring fingerÂ (CAT 1993)

Question 9
The product of all integers from 1 to 100 will have the following numbers of zeros at the end.
(a) 20
(b) 24
(c) 19
(d) 22Â (CAT 1993)

Question 10
Let x, y and z be distinct positive integers satisfying x < y < z and x + y + z = k. What is the smallest value of K that does not determine x, y, z uniquely?
(a) 9
(b) 6
(c) 7
(d) 8Â (CAT 1993)