Number Theory Previous Year Questions (CAT)  Set 0007

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

Question 1
The remainder, when (15^{23} + 23^{23}) is divided by 19, is
1. 4
2. 15
3. 0
4. 18 (CAT 2004)

Answer: Option 3

Question 2
Consider the sequence of a1, a2, a3 … to infinity where a_{1} = 81.33 and a_{2} = –19 and a_{j} = a_{j – 1} – a_{j – 2} for j > 3. What is the sum of the first 6002 terms of this sequence?
1. –100.33
2. –30.00
3. 62.33
4. 119.33 (CAT 2004)

Answer: Option 3

Question 3
The infinite sum 1 + 4/7 + 9/49 + 16/343 + …..equals
(1) 27/14
(2) 21/13
(3) 49/27
(4) 256/147 (CAT 2003)

Answer: Option 3

Question 4
Consider the sets T_{n}= {n, n + 1, n + 2, n + 3, n + 4), where n = 1, 2, 3, … , 96. How many of these sets contain 6 or any integral multiple thereof (i.e., any one of the numbers 6, 12, 18, …)?
1. 80
2. 81
3. 82
4. 83 (CAT 2003)

Answer: Option 1

Question 5
In a certain examination paper, there are n questions. For j = 1, 2, ..., n, there are 2^{(n − j)} students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is
(1) 12
(2) 11
(3) 10
(4) 9 (CAT 2003 Leaked)

Answer: Option 1

Question 6
What is the remainder when 4^{96} is divided by 6?
1. 0
2. 2
3. 3
4. 4 (CAT 2003)

Answer: Option 4

Question 7
If x and y are integers then the equation 5x + l9y = 64 has:
1. no solution for x < 300 and y < 0
2. no solution for x > 250 and y > –100
3. a solution for 250 < x < 300
4. a solution for –59 < y < –56 (CAT 2003)

Answer: Option 3

Question 8
Using only 2, 5, 10, 25 and 50 paise coins, what will be the minimum number of coins required to pay exactly 78 paise, 69 paise, and Re. 1.01 to three different persons?
1. 19
2. 20
3. 17
4. 1 (CAT 2003)

Answer: Option 1

Question 9
The seven basic symbols in a certain numeral system and their respective values are as follows:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur contiguously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the lefttoright reading occurs when a symbol is followed immediately by a symbol of greater value; then, the smaller value is subtracted from the larger.For example, XLVI = (50 – 10) + 5 + l = 46.
Q1) The value of the numeral MDCCLXXXVII is:
1. 1687
2. 1787
3. 1887
4.1987Q2) The value of the numeral MCMXCIX is:
1. 1999
2. 1899
3. 1989
4. 1889Q3) Which of the following can represent the numeral for 1995?
a. MCMLXXV
b. MCMXCV
c. MVD
d. MVM
1. only (a) and (b)
2. only (c) and (d)
3. only (b) and (d)
4. only (d) (CAT 2003)

Answer: Q1) Option 2 Q2) Option 1 Q3) Option 3

Question 10
What is the sum of all twodigit numbers that give a remainder of 3 when they are divided by 7?
1. 666
2. 676
3. 683
4. 777 (CAT 2003)