Number Theory Previous Year Questions (CAT)  Set 0013

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

Question 1
The remainder when 7^{84} is divided by 342 is
a. 0
b. 1
c. 49
d. 341 (CAT 1999)
The remainder when 7^{84} is divided by 342 is
a. 0
b. 1
c. 49
d. 341 (CAT 1999)

Question 2

Question 3
If n = 1+ x where x is the product of four consecutive positive integers, then which of the following
is/are true?
A. n is odd
B. n is prime
C. n is a perfect square
a. A and C only
b. A and B only
c. A only
d. None of these (CAT 1999)
If n = 1+ x where x is the product of four consecutive positive integers, then which of the following
is/are true?
A. n is odd
B. n is prime
C. n is a perfect square
a. A and C only
b. A and B only
c. A only
d. None of these (CAT 1999)

Question 4
If n^{2 }= 12345678987654321, what is n?
a. 12344321
b. 1235789
c. 111111111
d. 11111111Â (CAT 1999)
If n^{2 }= 12345678987654321, what is n?
a. 12344321
b. 1235789
c. 111111111
d. 11111111Â (CAT 1999)

Question 5
There are 50 integers a_{1}, a_{2}… a_{50}, not all of them necessarily different. Let the greatest integer of these 50 integers be referred to as G, and the smallest integer be referred to as L. The integers a_{1 }through a_{24 }form sequence S1, and the rest form sequence S2. Each member of S1 is less than or equal to each member of S2.
All values in S1 are changed in sign, while those in S2 remain unchanged. Which of the following statements is true?
a. Every member of S1 is greater than or equal to every member of S2.
b. G is in S1.
c. If all numbers originally in S1 and S2 had the same sign, then after the change of sign, the
largest number of S1 and S2 is in S1.
d. None of the above (CAT 1999)

Question 6
n^{3 } is odd. Which of the following statement(s) is(are) true?
I. n is odd.
II. n^{2 }is odd.
III. n^{2 }is even.
a. I only
b. II only
c. I and II
d. I and III (CAT 1998 )

Question 7
(BE)^{2} = MPB, where B, E, M and P are distinct integers. Then M =
a. 2
b. 3
c. 9
d. None of theseÂ (CAT 1998 )

Question 8
Fivedigit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?
a. 19800
b. 41976
c. 32976
d. None of these (CAT 1998 )

Question 9
A certain number, when divided by 899, leaves a remainder 63. Find the remainder when the same number is divided by 29.
a. 5
b. 4
c. 1
d. Cannot be determined (CAT 1998 )

Question 10
Three wheels can complete 60, 36 and 24 revolutions per minute. There is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch the ground again?
a. (5/2)s
b. (5/3)s
c. 6 s
d. 7.5 s (CAT 1998 )