On January 1, 2004 two new societies s1 and s2 are formed, each having n members. On the first day of each subsequent month, s1 adds b members while s2 multiples its current numbers 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?
Suppose n is an integer such that the sum of digits on n is 2, and 1010 < n < 1011. The number of different values of n is
The reminder, when (1523 + 2323) is divided by 19, is
If , then x when divided by 70 leaves a remainder of
Let n! = 1 × 2 × 3 × ... × n for integer . If p = 1! + (2 × 2!) + (3 × 3!) + ... + (10 × 10!), then
p + 2 when divided by 11!, Leaves a remainder of
The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?
Let S be the set of five-digit numbers formed by digits 1, 2, 3, 4 and 5, using each digit exactly once such that exactly two odd position are occupied by odd digits. What is the sum of the digits in the rightmost position of the numbers in S?
The rightmost non-zero digits of the number 302720 is
Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. the ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is
For a positive integer n, let pn denote the product of the digits of n and sn denote the sum of the digits of n. The number of integers between 10 and 1000 for which pn + sn = n is
Let S be a set of positive integers such that every element n of S satisfies the conditions
I.
II. every digit in n is odd
Then how many elements of S are divisible by 3?
Consider the set S = {2, 3, 4, ……, 2n + 1}, where 'n' is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X – Y?
How many pairs of positive integers m, n satisfy
,
where 'n' is an odd integer less than 60?
Let S be the set of all pairs (i, j) where, 1≤ i < j
≤ n and
n ≥ 4. Any two distinct members of S are called “friends” if they have one constituent of the pairs in common and “enemies” otherwise. For example, if
n = 4, then S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2) and (1, 3) are friends, (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.
For general ‘n’, how many enemies will each member of S have?
Let S be the set of all pairs (i, j) where, 1≤ i < j
≤ n and
n ≥ 4. Any two distinct members of S are called “friends” if they have one constituent of the pairs in common and “enemies” otherwise. For example, if
n = 4, then S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2) and (1, 3) are friends, (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.
For general ‘n’, consider any two members of S that are friends. How many other members of S will be common friends of both these members?
Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?
What are the last two digits of 72008?
Suppose, the seed of any positive integer n is defined as follows:
seed(n) = n, if n < 10
= seed(s(n)), otherwise,
where s(n) indicates the sum of digits of n. For example,
seed(7) = 7, seed(248) = seed(2 + 4 + 8) = seed(14) = seed (1 + 4) = seed (5) = 5 etc. How many positive integers n, such that n < 500, will have seed (n) = 9?
Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?
273 − 272 − 271 is the same as
The number of integers n satisfying −n + 2 ≥ 0 and 2n ≥ 4 is
A square piece of cardboard of sides ten inches is taken and four equal squares pieces are removed at the corners, such that the side of this square piece is also an integer value. The sides are then turned up to form an open box. Then the maximum volume such a box can have is
x, y and z are three positive integers such that x > y > z. Which of the following is closest to the product xyz?
What is the greatest power of 5 which can divide 80! exactly.
A third standard teacher gave a simple multiplication exercise to the kids. But one kid reversed the digits of both the numbers and carried out the multiplication and found that the product was exactly the same as the one expected by the teacher. Only one of the following pairs of numbers will fit in the description of the exercise. Which one is that?
Find the minimum integral value of n such that the division 55n/124 leaves no remainder.
Let k be a positive integer such that k + 4 is divisible by 7. Then the smallest positive integer n, greater than 2, such that k + 2n is divisible by 7 equals
A calculator has two memory buttons, A and B. Value 1 is initially stored in both memory locations. The following sequence of steps is carried out five times:
add 1 to B
multiply A to B
store the result in A
What is the value stored in memory location A after this procedure?
If x is a positive integer such that 2x + 12 is perfectly divisible by x, then the number of possible values of x is
A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p2 − 1) is
To decide whether a n digits number is divisible by 7, we can define a process by which its magnitude
is reduced as follows: (i1, i2, i3,
, are the digits of the number, starting from the most significant
digit). i1 i2
. in ⇒ i1. 3n-1 + 12 . 3n-2 +
+ in . 30.
E.g. 259 ⇒ 2.32 + 5.31 + 9.30 = 18 + 15 + 9 = 42
Ultimately the resulting number will be seven after repeating the above process a certain number of times. After how many such stages, does the number 203 reduce to 7?
If 8 + 12 = 2, 7 + 14 = 3, then 10 + 18 = ?
An intelligence agency decides on a code of 2 digits selected from 0, 1, 2, .... , 9. But the slip on which the code is handwritten allows confusion between top and bottom, because these are indistinguishable. Thus, for example, the code 91 could be confused with 16. How many codes are there such that there is no possibility of any confusion?
Suppose one wishes to find distinct positive integers x, y such that (x + y)/ xy is also a positive integer. Identify the correct alternative.
Given odd positive integers x, y and z, which of the following is not necessarily true?
139 persons have signed up for an elimination tournament. All players are to be paired up for the first round, but because 139 is an odd number one player gets a bye, which promotes him to the second round, without actually playing in the first round. The pairing continues on the next round, with a bye to any player left over. If the schedule is planned so that a minimum number of matches is required to determine the champion, the number of matches which must be played is
The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is
Let x < 0, 0 < y < 1, z > 1. Which of the following may be false?
A young girl counted in the following way on the fingers of her left hand. She started calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, index finger 8, thumb 9, then back to the index finger for 10, middle finger for 11, and so on. She counted up to 1994. She ended on her
The product of all integers from 1 to 100 will have the following numbers of zeros at the end.
An emotionally honest learning space can only be created by
Conceptual space with words can be created by
The author argues that the Japanese system
The growth of popularity of business schools among students was most probably due to
According to the passage
A criticism that management education did not face was that
The absence of business schools in Japan
The 1960s and 1970s can best be described as a period
US business schools faced criticism in the 1980s because
Training programmes in Japanese corporations have
The Japanese modified their views on management education because of
The Japanese were initially able to do without business schools as a result of
The main difference between US and Japanese corporations is
The author argues that
What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?
Direction for questions 51 to 53: Answer these questions independently.
56 1 is divisible by
Direction for questions 58 to 87: Answer the questions independently.
72 hens cost Rs.__ 96.7__. Then what does each hen cost, where two digits in place of __ are not
visible or are written in illegible hand?
Direction for questions 58 to 87: Answer the questions independently.
Direction for questions 58 to 87: Answer the questions independently.
For the product n(n + 1)(2n + 1), n ∈ N, which one of the following is not necessarily true?
Direction for questions 58 to 87: Answer the questions independently.
The remainder obtained when a prime number greater than 6 is divided by 6 is
Direction for questions 58 to 87: Answer the questions independently.
Three consecutive positive even numbers are such that thrice the first number exceeds double the third
by 2, then the third number is
Direction for questions 58 to 87: Answer the questions independently.
Three bells chime at an interval of 18 min, 24 min and 32 min. At a certain time they begin to chime
together. What length of time will elapse before they chime together again?
Direction for questions 58 to 87: Answer the questions independently.
What is the value of m which satisfies 3m2 21m + 30 < 0?
If a number 774958A96B is to be divisible by 8 and 9, the respective values of A and B will be
If n is any odd number greater than 1, then n(n2 1) is
Once I had been to the post office to buy five-rupee, two-rupee and one-rupee stamps. I paid the clerk Rs. 20, and since he had no change, he gave me three more one-rupee stamps. If the number of stamps of each type that I had ordered initially was more than one, what was the total number of stamps that I bought?
Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?
A red light flashes three times per minute and a green light flashes five times in 2 min at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour?
Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The number of boxes containing the same number of oranges is at least
In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?
Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?
m is the smallest positive integer such that for any integer n ≥ m, the quantity n3 7n2 + 11n 5 is positive. What is the value of m?
Let b be a positive integer and a = b2 b. If b ≥ 4 , then a2 2a is divisible by
Ashish is given Rs. 158 in one-rupee denominations. He has been asked to allocate them into a number of bags such that any amount required between Re 1 and Rs. 158 can be given by handing out a certain number of bags without opening them. What is the minimum number of bags required?
Let n be the number of different five-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?
How much does R owe to S in Thai Baht?
How much does M owe to S in US Dollars?
A test has 50 questions. A student scores 1 mark for a correct answer, 1/3 for a wrong answer, and 1/6 for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student cannot be less than
How many even integers n, where 100 ≤ n ≤ 200 , are divisible neither by seven nor by nine?
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
How many three digit positive integers, with digits x, y and z in the hundred's, ten's and unit's place respectively, exist such that x < y, z < y and x ≠ 0 ?
The number of positive integers n in the range 12 ≤ n ≤ 40 such that the product (n −1)(n − 2)...3.2.1 is not divisible by n is
What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7?
What is the remainder when 496 is divided by 6?
Let n ( >1) be a composite integer such that is not an integer. Consider the following statements:
A: n has a perfect integer-valued divisor which is greater than 1 and less than
B: n has a perfect integer-valued divisor which is greater than but less than n
If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is
Let a, b, c, d and e be integers such that a = 6b = 12c, and 2b = 9d = 12 e. Then which of the following pairs contains a number that is not an integer?
Let x and y be positive integers such that x is prime and y is composite. Then,
The remainder when 260 is divided by 5 equals
Mr.X enters a positive integer Y in an electronic calculator and then goes on pressing the square – root key repeatedly. Then
If n is any positive integer, then n3 – n is divisible
If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is
While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is
The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157:3, then the sum of the two numbers is
If 5x – 3y = 13438 and 5x–1 + 3y+1 = 9686, then x + y equals
How many pairs (a, b) of positive integers are there such that a ≤ b and ab = 42017?
A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is
The number of all natural numbers up to 1000 with non-repeating digits is