Number Theory Previous Year Questions (CAT)  Set 0006

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

Question 1
Each family in a locality has at most two adults, and no family has fewer than 3 children. Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families. Then the minimum possible number of families in the locality is
1. 4
2. 5
3. 2
4. 3 (CAT 2004)

Answer: Option 4

Question 2
n is an integer such that the sum of the digits of n is 2, and 10^{10} < n < 10^{11}. The number of different values for n is
1. 11
2. 10
3. 9
4. 8 (CAT 2004)

Answer: Option 1

Question 3
In Nuts and Bolts factory, one machine produces only nuts at the rate of 100 nuts per minute and needs to be cleaned for 5 minutes after production of every 1000 nuts. Another machine produces only bolts at the rate of 75 bolts per minute and needs to be cleaned for 10 minutes after production of every 1500 bolts. If both the machines start production at the same time, what is the minimum duration required for producing 9000 pairs of nuts and bolts?
1. 130 minutes
2. 135 minutes
3. 170 minutes
4. 180 minutes (CAT 2004)

Answer: Option 3

Question 4
On January 1, 2004 two new societies, S1 and S2, are formed, each with n members. On the first day of each subsequent month, S1 adds b members while S2 multiplies its current number of members by a constant factor r. Both the societies have the same number of members on July 2, 2004. If b = 10.5n, what is the value of r?
1. 2.0
2. 9
3. 1.8
4. 1.7 (CAT 2004)

Answer: Option 1

Question 5
If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?
1. 0
2. –1
3. 1
4. Not unique (CAT 2004)

Answer: Option 1

Question 6
If a/(b + c) = b/(a + c) = c/(a + b) = r, then r cannot take any value except
(1) 1/2
(2) 1
(3) 1/2 or 1
(4) 1/2 or 1 (CAT 2004)

Answer: Option 3

Question 7
A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals
(1) 31
(2) 63
(3) 75
(4) 91 (CAT 2003 Leaked)

Answer: Option 4

Question 8
There are 8436 steel balls, each with a radius of 1 centimetre, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. The number of horizontal layers in the pile is
(1) 34
(2) 38
(3) 36
(4) 32 (CAT 2003 Leaked)

Answer: Option 3

Question 9
N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a twominute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?
1. 5
2. 7
3. 9
4. None of the above (CAT 2004)

Answer: Option 2

Question 10