Number Theory Previous Year Questions (CAT)  Set 0012

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

Question 1
Sam has forgotten his friend’s sevendigit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number nine appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?
(1) 1000
(2) 2430
(3) 3402
(4) 3006 (CAT 2000)
Sam has forgotten his friend’s sevendigit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number nine appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?
(1) 1000
(2) 2430
(3) 3402
(4) 3006 (CAT 2000)

Question 2
Let N = 55^{3 }+ 17^{3 }– 72^{3}. N is divisible by
(1) both 7 and 13
(2) both 3 and 13
(3) both 17 and 7
(4) both 3 and 17 (CAT 2000)
Let N = 55^{3 }+ 17^{3 }– 72^{3}. N is divisible by
(1) both 7 and 13
(2) both 3 and 13
(3) both 17 and 7
(4) both 3 and 17 (CAT 2000)

Question 3
If x^{2}+ y^{2}= 0.1 and x – y = 0.2, then  x  +  y  is equal to
(1) 0.3
(2) 0.4
(3) 0.2
(4) 0.6 (CAT 2000)

Question 4
A shipping clerk has five boxes of different but unknown weights each weighing less than 100 kg. The clerk weighs the boxes in pairs. The weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121 kg. What is the weight, in kg, of the heaviest box?
(1) 60
(2) 62
(3) 64
(4) Cannot be determined (CAT 2000)
A shipping clerk has five boxes of different but unknown weights each weighing less than 100 kg. The clerk weighs the boxes in pairs. The weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121 kg. What is the weight, in kg, of the heaviest box?
(1) 60
(2) 62
(3) 64
(4) Cannot be determined (CAT 2000)

Question 5
Convert the number 1982 from base 10 to base 12. The result is
(1) 1182
(2) 1912
(3) 1192
(4) 1292 (CAT 2000)
Convert the number 1982 from base 10 to base 12. The result is
(1) 1182
(2) 1912
(3) 1192
(4) 1292 (CAT 2000)

Question 6
A farmer has decided to build a wire fence along one straight side of his property. For this, he planned to place several fenceposts at six metre intervals, with posts fixed at both ends of the side. After he bought the posts and wire, he found that the number of posts he had bought was five less than required. However, he discovered that the number of posts he had bought would be just sufficient if he spaced them eight metres apart. What is the length of the side of his property and how many posts did he buy?
(1) 100 metres, 15
(2) 100 metres, 16
(3) 120 metres, 15
(4) 120 metres, 16 Â Â Â (CAT 2000)

Question 7
Consider a sequence of seven consecutive integers. The average of the first five integers is n. The average of all the seven integers is
(1) n
(2) n + 1
(3) K × n, where K is a function of n
(4) n + 2/7 (CAT 2000)
Consider a sequence of seven consecutive integers. The average of the first five integers is n. The average of all the seven integers is
(1) n
(2) n + 1
(3) K × n, where K is a function of n
(4) n + 2/7 (CAT 2000)

Question 8
There is a vertical stack of books marked 1, 2, and 3 on TableA, with 1 at the bottom and 3 on top. These are to be placed vertically on TableB with 1 at the bottom and 2 on the top, by making a series of moves from one table to the other. During a move, the topmost book, or the topmost two books, or all the three, can be moved from one of the tables to the other. If there are any books on the other table, the stack being transferred should be placed on top of the existing books, without changing the order of books in the stack that is being moved in that move. If there are no books on the other table, the stack is simply placed on the other table without disturbing the order of books in it. What is the minimum number of moves in which the above task can be accomplished?
(1) One
(2) Two
(3) Three
(4) Four (CAT 2000)

Question 9
The number of positive integer valued pairs (x, y) satisfying 4x – 17y = 1 and x ≤1000 is
a. 59
b. 57
c. 55
d. 58 (CAT 1999)
The number of positive integer valued pairs (x, y) satisfying 4x – 17y = 1 and x ≤1000 is
a. 59
b. 57
c. 55
d. 58 (CAT 1999)

Question 10
Let a, b, c be distinct digits. Consider a twodigit number ‘ab’ and a threedigit number ‘ccb’, both defined under the usual decimal number system, if (ab)^{2}= ccb > 300, then the value of b is
a. 1
b. 0
c. 5
d. 6 (CAT 1999)
Let a, b, c be distinct digits. Consider a twodigit number ‘ab’ and a threedigit number ‘ccb’, both defined under the usual decimal number system, if (ab)^{2}= ccb > 300, then the value of b is
a. 1
b. 0
c. 5
d. 6 (CAT 1999)