Question Bank  Number Theory  Shashank Prabhu, CAT 100 Percentiler

Q73) In a box containing 15 apples, exactly 6 apples are rotten. Each day one apple is taken out from the box. What is the probability that after four days there are exactly 8 apples in the box that are not rotten?
a. 12/91
b. 1/7
c. 2/13
d. None of these[OA: Option a]

Q74) What is the remainder when n! + (n! + 1) + (n! – 2) + (n! + 3) ..... + (n! – 2006) is divided by 1003 for n = 1003?
[OA: 0]

Q75) What is the 10th positive integer that cannot be expressed as the sum of two or more consecutive positive integers?
[OA: 512]

Q76) What is the 2037th positive integer that can be expressed as the sum of two or more consecutive positive integers?
[OA: 2049]

Q77) No. of positive integral solutions for 3/x  7/y = 1/14
[OA: 10]

42y  98x = xy
y(42  x) + 98(42  x) = 4116
(42  x)(98 + y) = 4116
x has to be less than 42. So there are 10 values of x that are possible.

Q78) Sanjay has exactly six sealed boxes containing 15, 31, 19, 20, 16 and 18 coins. Out of the six boxes with Sanjay, there are exactly five boxes that contained silver coins whereas one box contained gold coins. He distributed all the six boxes among his three sons in such a manner that his eldest son got the only box with gold coins and the other boxes were distributed in such a manner so that other two brothers received the silver coins in the ratio of 2:1. How many gold coins were there in one of the boxes with Sanjay? (Assume no coins were taken out of the boxes)
[OA: 20]

Q79) Tania prepares for the CAT examination by practicing for 100 days. On any of these 100 days she does not solve more than 20 questions. If on any day, she solves more than 12 questions, then she solves at most 6 questions each on the next two days. What is the maximum possible number of questions that she can solve over the period of 100 days?
[OA: 1208]

odd  even should be 0/11/22...
We know that the total is 45. So, odd  even should be 11.
The two sets are 28 and 17. 28 can be formed by using 9874, 9865. 17 can be formed by using 9521, 9431, 8621, 8531, 8432, 7631, 7541, 7532, 6542. So 11 cases in total. The digits can be arranged in 4!5! ways. Total 11*4!*5!

Q80) A is a set of all those integers greater than 1 and less than 100 which are divisible either by 3 or by 4
but not by both. What is the index of the highest power of thousand that occurs in the product of all the elements Of set A?
a) 9
b) 7
c) 3
d) 4
e) 6[OA: 3]

Q81) if (y + z  x)/x, (x + z  y)/y and (x + y  z)/z are in AP, then which of the following is also in AP?
a) x, y, z
b) x + y, x + z, y + z
c) 1/x, 1/y, 1/z
d) 1/(x+y), 1/(x+z), 1/(y+z)
e) None of the above[OA: Option c]

Q82) Himanshu, who wanted to try out his new car started from his house at 9:00 am and traveled at a uniform speed of 25 km/hr to reach a point A. From A, he travelled to another point B at a uniform speed of 30 km/hr. He then returned to A at a uniform speed of 60 km/hr and from A, he returned to his house at a uniform speed of 100 km/hr and reached his house at 12:00 noon. What is the total distance (in km) travelled by Himanshu?
a) 75
b) 300
c) 120
d) Cannot be determined[OA: Option c]

@shashank_prabhu 1.4
1/2x5 = 1/234
X=2.4
Distance =(2.41) = 1.4

@shashank_prabhu let a=b=c = gongs of tiger group
Tiger = 3a
N= total number
Tiger = N/2 = 3a
N = 6a
Now we have a=18
N = 6*18 = 108

Q86) All the digits from 1 to 9 are written in line on a blackboard. Henry erases two of the digits and appends the digits in the product of the digits he has erased at the end of the previous series of digits (for example initial series of digits: 1 2 3 ......9. If the two digits erased are 5 and 9 (product: 45), digits in the product of these two digits i.e. 4 and 5 are appended at the end to get new series: 1 2 3 4 6 7 8 4 5). When only two digits remain on the blackboard, they are replaced by the digits in the product of those two digits. This process is repeated over and over again to obtain only one digit in the end. What is the digit remaining on the blackboard in the end?
[OA: 0]

Q87) A cat, which is sitting inside a tunnel PQ, at a distance of 50 m from the end P, notices a train approaching the end P of the tunnel from the outside. Now, if the cat runs towards the end P, then the train would meet it exactly at P. If the cat runs towards the end Q instead, then the train would meet it exactly at Q. Which of the following is not a possible value of the length PQ (in m) of the tunnel?
a) 130
b) 100
c) 120
d) 110
e) 105[OA: Option b]

Q88)
[OA: 3]

Q89) If a cuboid of dimension 60 × 40 × 30 cm^3 is cut into smaller cuboids of integral dimension having shape similar to the original cuboid, then which of the following cannot be the number of smaller cuboids ?
a. 1000
b. 125
c. 8
d. 64
e. None of these[OA: 64]

Q90) The number of terms common in the two sequences 2, 6, 12, 20, ..., 930 and 4, 8, 12, 16 .... 960 is
[OA: 14]

Q93) The digits of all the two digit multiples of 8 are reversed. How many of the resulting numbers would be divisible by 4 but not by 8?
[OA: 2]