QBM002
M151. What is the total number of different divisors including 1 and the number that can divide the number 3600?
M152. A set X consists of 100 natural nos, each of which is a perfect cube. The maximum no of elements of X that one can always find such that each of them leaves the same remainder when divided by 17 is
M153. Let x and y be positive whole numbers, and let p be any odd prime.
It is well known that x3 + y3 is never equal to an odd prime.
But given that n is a positive integer which contains an odd factor greater than one, prove that xn + yn = p has no solutions.
M154. The largest number amongst the following that will perfectly divide 101100 - 1 is
(1) 100 (2) 10,000 (3) 100100 (4) 100,000
M155. (x, y) is a pair of non-negative integers such that (x+ y - 5)^2 = 9xy.
What are the possible values of (x,y)
QM156.
QM157. How many four digit numbers exist which can be formed by using the digits 2, 3, 5 and 7 once only such that they are divisible by 25?
M158. A positive integer, n, is divided by d and the quotient and remainder are q and r respectively. In addition d, q, and r are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.
M159. Given that x and y are integer, how many different solutions does the equation |x| + 2|y| = 100 have?
M160. A number consisting entirely of the digit one is called a repunit; for example, 11111.
Find the smallest repunit that is divisible by 63.
END
QBM001
M1. Which is greater of the two 2300 or 3200
M2. Find the last two digits of 15 x 37 x 63 x 51 x 97 x 17 ? Ans. 35
M3. If both 112 and 33 are factors of the number x * 43 * 62 * 1311, then what is the smallest possible value of 'x'?
M4.How many numbers are in between 140 and 259(both included) which are divisible by 7?
M5.11 is the smallest prime made up of the same digit.
M6.Find the number of numbers between 300 and 400(both included) that are not divisible by 2, 3, 4 and 5.
M7. Find the sum of all three-digit numbers that give a remainder of 3 when they are divided by 7. 676
M8. Explain why a number made up of the same digit can only be prime if the repeating digit is one AND the number of digits is itself prime.
M9.A four digit number A375 is divisible by 11. Find A?
M10.Find the sum of all odd three digit numbers that are divisible by 5?
M11.Given that n
2 and an 1 is prime, prove that a = 2, and n must be prime
M12. Which is greater of the two 1113 or 1311
M13. A five-digit number A290B is divisible by 11 and 13. Find A and B?
M14. A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?
M15. How many two digit primes can you find for which their reverse is also prime?
END
QBM003
M16. The product of a two-digit number by a number consisting of the same digits written in the reverse order is equal to 2430. Find the lower number? 45
M17. For each of the numbers: 41, 83, 32, the first digit is greater in value than the second digit.How many 2-digit numbers have this property?
M18. Two numbers when divided by a certain divisor leave remainders of 431 and 379 respectively. When the sum of these two numbers is divided by the same divisor, the remainder is 211. What is the divisor?
M19.How many 3-digit numbers exist for which the sum of the digits is six?
M20. Find the smallest number, greater than 1, which has a remainder of 1 when divided by any of 2, 3, 4, 5, 6, or 7
M21. 165 + 215 is divisible by 33 , 13 , 27 or 31?
M22. How many pairs of natural numbers are there the difference of whose squares is 45?
M23. Given that a, b, and c are natural numbers, solve the following equation. a!b! = a! + b! + c2
M24. A five digit number 3A25B is divisible by 19 and 7. Find A and B?
M25. How many keystrokes are needed to type numbers from 1 to 100?
M26. How many 3-digit numbers have two digits the same?
M27. 14n + 11 can never be a
a. Prime b. odd c.even
M28. The remainder obtained when 43101 + 23101 is divided by 66?
M29. What is the least number that should be multiplied to 100! to make it perfectly divisible by 350
M30. How many times will the digit '0' appear between 1 and 10,000?
END
QBM005
M31. If Sn represents the sum of the first n odd numbers, then 4Sn?
a. S2n b. S4n c. 2S2n d. ( S2n)2
M32. Given that p is prime, when is 8p+1 square?
M33. A tortoise and a hare race against each other. A hare runs at a constant speed of 36 km per hour for exactly ten seconds and waits for the tortoise to catch up. If the tortoise takes two hours to move 1 km, how long will it take to catch up?
M34. 22225555 + 555552222 is divided by 7, the remainder is
M35. When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the divisor?
M36. Consider the infinite sequence:, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, ... What is the 1000th term?
M37. 32n-1 is divisible by 2n+3 for n= ?
M38. a!b! = a! + b! + 2c
M39. Find the value of the digit c in the following calculation ab - ba = c4
M.40 (323232)/9 will leave a remainder
M41. Find the sum of all even three digit numbers that are divisible by 8 and 10?
M42. A bag contains n discs, made up of red and blue colours. Two discs are removed from the bag.If the probability of selecting two discs of the same colour is 1/2, what can you say about the number of discs in the bag?
M43. Find the remainder when that the number 1989 x 1990 x 19923 gives when divided by 3.
M44. Find the greatest number of five digits, which is exactly divisible by 7, 10, 15, 21 and 28.
Q45. Four digits are selected from the set {1,2,3,4,5} to form a 4-digit number. Find the sum of all possible permutations
END
QBM006
M46. One pendulum ticks 57 times in 58 seconds and another 608 times in 607 seconds. If they start simultaneously, find the time after which they will tick together?
M47. Find the last digit of the number 12 + 22 + 32 + . . . . . . . 992
M48. Prove that 10n-1 is divisible by 11 if n is even and 10n+1 is divisible by 11 if n is odd.
M49. Find the greatest number of five digits, which is exactly divisible by 7, 10, 3, 11 and 28.
M50. Find the remainder when 1010+ 10100+ 101000 + . . . +1010000000000 is divided by 7.
M51. 2! + 4! + 6! + 8! + . . . + 100! When divided by 3 would leave the remainder
M52. 6n + 8n is divisible by 7 iff n is odd. Even integer prime
M3. How many numbers are there between 500 and 600 in which 9 occurs only once?
M54. Let n be the number of different 5 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?
M55. If x = 174 and y= 14 x 16 x 18 x 20 , then which of the following is true
x >y , x < y , x =y
M56. How many different pairs of natural numbers add to make one-thousand?
M57. Which among the following is greatest √7 + √3 , √5 + √5 , √6 + 2?
M58. In deciding who should pay for lunch, Jane challenges John and James to a game of chance, "I shall take two ordinary 6-sided dice, roll them, and add their scores together. Then one of you shall do the same. If the second total is higher, John pays for lunch, if it is lower, James pays, or if it is same, I will pay. As there are three equally likely outcomes, the game is fair." Is the game fair?
M59. 461 + 462 + 463 + 464 + 465 4 is divisible by 3 , 5, 11 or 17.
M60. A 6 digit number 200ABC is divisible by 13, 11 and 7. Find A, B and C?
END
QBM007
M61. How many cuboids exist for which the volume is less than 100 units3 and the integer side lengths are in an arithmetic sequence?
M62. The number formed by writing any digit 6 times is always divisible by 1001 , 7 , 13 or all of these
M63. Find the exact value of the following infinite series:
1/2! + 2/3! + 3/4! + 4/5! + ...
M64. How many digits are used to make a book containing 500 pages?
M65. The sum of the digits of a two digit number is 5. If we put the digits of the numbers in reverse order, the new number s 41 less than the original number. Find 33% of the number?
M66. In a particular class, each student has blonde hair or brown eyes; 1/4 of students with blonde hair have brown eyes and 1/3 of the students with brown eyes have blonde hair. What fraction of the class in total have brown eyes?
M67. Sushil had to do a multiplication. Instead of taking 35 as one of the multipliers, he took 53. As a result, the product went up by 540. What is the new product?
M68. How many primes less than 100 can be written as the sum of two square numbers?
M69. How many zeroes will be there at the end of 1 x 2 x 3 x . . . . . . x 1000 ?
M70. Given that p is prime, when is 4p + p4 prime?
M71. How many digits does the number 21000 contain?
M72.How many zeroes will be there at the end of 11 x 22 x 33x . . . . . . x 100100 ?
M73. When travelling by aircraft, passengers have a maximum allowable weight for their luggage. They are then charged ?10 for every kilogram overweight. If a passenger carrying 40 kg of luggage is charged ?50, how much would a passenger carrying 80 kg be charged?
M74. Mr. and Mrs. Roberts have two daughters and three sons. At Easter time every member of the family buys one chocolate Easter egg for each other member. How many Easter eggs will be bought in total?
M75. What is the least number which has no remainder when divided by any number from 1 to 10?
END
QBM008
M76. How many zeroes will be there at the end of 12 x 22 x 32 x . . . . . . x 1002 ?
M77. In my pocket I have exactly 15p, comprising of eight coins made up of 1p, 2p and 5p pieces. How many of each coin do I have?
M78. When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?
M79. If every combination of the digits 1,2,3,4 was written down, what would be the sum of the numbers formed?
M80. Find the sum of all even four digit numbers that are divisible by first five natural numbers in number system?
M81. If F(x)= sum of all the digits of x, where x is a natural number, then what is the value of
F(101) + F(102) + F(103). . . . + F(200)
M82. A box contains a mixture of black and white discs, with more black than white discs. A game is played by taking one disc at a time, at random, and without replacement. If an equal number of each colour have been removed the game stops and the player wins. It is found that the player has an equal chance of winning or losing. If the box contains twelve discs in total, find the number of black discs
M83. If F(x)= sum of all the digits of x, where x is a natural number, then what is the value of
F2(101) + F2(102) + F2(103). . . . + F2(110)
M84. During a football match in a tournament a goalkeepers save rate was given as 33%. After saving the next shot on target it rose to 40%. How many more shots on target does he need to save to raise his save rate to 50%?
M85. The number of pairs of consecutive positive integers whose product is less than 2000 is?
What is the highest power of 31 in 1000!?
M86. A train, length 1/2 mile, travelling at a constant speed of 30 miles per hour enters a tunnel which is 2 miles long. How long will it take for the train to completely pass through the tunnel?
M87.How many numbers in the first 100 natural numbers are not divisible by 2, 3 or 5.
M88. When a number is divided by 29, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?
M89. A train travels a distance of 90 miles from A to B in one hour. Another train sets off at the same time and travels from B to A, taking two hours to complete the journey.How many miles from A did the two trains cross?
M90. How many zeros will be there at the end of 37! ?
END
QBM009
M91. A girl bought 15 pens costing ?1.84. She paid one pence more for each red pen than each blue pen. How many of each kind did she buy and at what price?
M92. Consider the following series.
S(n) = [1/(√1+√2) ] + [1/(√3+√2) ] + [1/(√3+√4) ]+ . . + [1/(√(n+1)+√n) ]
For which values of n is S(n) rational?
M93. the product of three consecutive integers, plus their mean, is always a. cube b. square c. both
M94. How many zeros will be there at the end of (2!)2! x (4!)4! x (6!)6! x (8!)8!
M95. What is the minimum number of square marbles required to tile a floor of length 5 metres 78 cm and width 3 metres 74 cm?
M96. When I was 14 years old my father was 42 years old, which was three times my age. Now he is twice my age, how old am I?
M97. What is the remainder when N is divided by 37 where N = 3636 + 36.
M98. There are six simple fractions involving ninths that cannot be cancelled down. How many simple fractions with a denominator equal to 24 cannot be cancelled down?
M99. What proportion of 3-digit numbers contain the digit one?
M100. Matilda's father takes 20 minutes to mow the back garden lawn, but Matilda takes 30 minutes to do the same job. If they worked together, how long would it take to cut the lawn?
M101. N = 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + . . . . . . + 1/156, the value of N is
12/13 13/12 1/13 None
M102. Convert 423 from base 5 to base 8
M103. A person starts multiplying consecutive positive integers from 11. How many numbers should he multiply before the will have result that will end with 3 zeroes?
M104. How many 2-digit numbers have an odd product? (i.e the product of the 2-dgit numbers is an odd number)
M105. Convert 671 from base 9 to base 3
END
QBM010
M106. For what value of 'n' will the remainder of 351n and 352n be the same when divided by 7?
M107. If a is real then the minimum value of a2 - 8a + 11 is ________?
M108. Prove that for any natural number, n, there exists a sequence of (n 1) consecutive numbers that are composite.
M109. Convert 101001001 from base 2 to base 5
M110. Prove that 99n ends in 99 for odd n
M111.When Christmas trees are planted they should stand at least 2 metres away from one another whilst growing.What is the maximum number of trees that can be planted in one square kilometre?
M112. Convert 54321 from base 6 to base 10
M113. A wooden cube has each of its six faces painted either white or grey. How many different cubes can be made out of it?
M114.96 + 1 when divided by 8, would leave a remainder
M115. How many numbers below one hundred are divisible by both 2 and 3?
M116. For what value of p , 2p + 3p can be a perfect square.
Ans: never
M117. Find the largest number which when divides 77, 147 and 252 leaves the same remainder
a. 50 b. 35 c. 45 d. 30
M118. Prove that seven is the only prime number that is one less than a perfect cube.
M119. What smallest number should multiply 34300, in order to make it a perfect square?
M120. 3p + 3q = 2430 and it is known that p & q are positive then p + q equals
a. 4 b. 10 c. 12 d. 8
END
QBM011
M121.
M122. Twenty-seven small red cubes are connected together to make a larger cube that measures 3 x 3 x 3. All of its external faces are painted white and the cube is dismantled.
M123. Find the remainder of 12345678987654321 divided by 328 ?
M124. In order to win a game of darts, a player must finish on exactly zero and their last dart must land in a double or hit the bull's eye.For example, if a player hit treble twenty (60 points), double twenty (40 points) and the bull's eye (50 points) they would score 150 points and could use this combination to finish.How many ways can a player finish from 150 points with three darts?
M125. In How many ways can a number 6084 be written as a product of two different factors?
M126. The 7th digit of (202)3 is
M127. Can you prove that (2n)! is divisible by 22n − 1?
M128. Sum of all two digit numbers which are divisible by 9?
M129. If the number 3402 is converted from base 10 to base x, it becomes 12630. what is the value of x ?
M130. How many of the numbers under 100 that are divisible by 10 can you make by adding four consecutive integers?
M131. Two railway stations, P and Q, are 279 miles apart. A train departs from P at 2pm and travels at a constant speed of 51 mph towards Q.
At 3pm a second train begins a journey from Q towards P at a constant speed of 60 mph.How far apart are the two trains twenty minutes before they pass each other?
M132. Can you prove that the expression 21n − 5n + 8n is divisible by 24?
M133. Divide 136 into 2 parts such that when one divided by 8 leaves the remainder of 3 and other divided by 5 it leave the remainder of 2. How many such sets are there
a. 3 b. 2 c. 4 d.1
M134. abc is a three-digit natural number so that abc = a! + b! + c!. what is the value of (b + c)a ?
M135. Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.
END
QBM012
M136. Find the value of 1.1! + 2.2! + 3.3! + ......+n.n!
M137. What will be the remainder when 25625 + 26 is divided by 247 ?
M138. The ratio of boys to girls at a school disco is 9:10. An extra 17 boys arrive and the ratio changes to 8:7.How many girls are there at the disco?
M139. 461 + 462 +463 +464 is divisible by
a. 18 b. 17 c. 20 d. None
M140. If an n-digit natural number is added to a number made by putting the digits of the original number in reverse order the sum is always divisible by k where n is an even number, then k must be a multiple of
22 111 11 None
M141. Given that [n(n+1)(n+2)]2 = 3039162537*6, find the value of *.
M142. By what least number must 217800 be multiplied in order to make it a perfect square?
M143. What is the highest power of 82 contained in 83! - 82! ?
M144. What is the value of M and N respectively? If M39048458N is divisible by 8 & 11; Where M & N are single digit integers
M145. How many numbers are there between 200 and 400, which are divisible by 11 and 3, but not by 2?
M146. 32n - 1 is divisible by 2n+3 for n = ? 2 , 3 , 4 or none
M147. If you had discs numbered 1 to 10, how would you separate the discs into the two bags such that no bag contains its double?
M148. What is the smallest 4 digit number which leaves the remainder 2 when divided by 6 or 7 but is exactly divisible by 11.
M149. A five digit number 3A25B is divisible by 19 and 7. Find A and B?
M150. If N = 2 x 4 x 6 x 8 . . . 100 How many zeroes are there at the end of N?
END
QBM045
Q1. The number FIVE as written using block capitals contains exactly 10 strokes or segments of a straight line. Find a number which when written out in words (using no tricks) contains as many strokes as the number says.
Q2. The number of 1's in the binary notation of 289 - 1 is
(a) 89
(b) 88
(c) 90
(d) 1
Q3. The highest power of 2 in 10! + 11! + 12! + 13! + ...+ 1000! is
(a) 8
(b) 9
(c) 10
(d) 11
Q4. Assume that all bricklayers work at the same rate of speed. If it takes nine bricklayers (all working at the same time) fourteen days to do a job, how long would it take for the job to be done by
a) seven bricklayers
b) three bricklayers.
Q5. If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of a?
1. 121
2. 3267
3. 363
4. 33
Q6. Let (x, y) be co-prime numbers. Then
(a) x + y and x - y have no common factor other than 1 and 3 for all values of x and y.
(b) x + y and x - y have no common factor other than 1 and 2 for all values of x and y.
(c) x + y and x - y have no common factor other than 1 for all values of x and y.
(d) none of the above
Q7. What is the least number that should be multiplied to 100! To make it perfectly divisible by 718
a. 1
b. 7
c. 21
d. 49
Q8. A car has traveled 24,000 km and, in that distance, has worn out 6 tyres. Each tyre travelled the same distance. How far did each separate tyre travel?
Q9. The remainder when 3256 is divided by 100 is
(a) 61
(b) 21
(c) 41
(d) 81
Q10. Find the sum 1/(1.2) + 1/(2.3) + . . . .+ 1/(10.11)
a. 1
b. 2
c. 10/11
d. 11/12
e. 2531/2520
Q11. Find the greatest number of five digits, which is exactly divisible by 7, 10, 15, 21 and 28.
(1) 99840
(2) 99900
(3) 99960
(4) 99990
Q12. The number of even factors of the number N = 1233 x 344 x 522, is
(a) 144
(b) 168
(c) 24
(d) 7
Q13. A person had to multiply two numbers. Instead of multiplying by 12, the person multiplied by 21, and the product went up by 270. What was the final product?
a. 252
b. 360
c. 630
d. None of these
End
QBM046
Q1. An old man has Rs (1! + 2! + 3! + ...+ 50!), all of which he wants to divide equally (without fractions) among his n children. Then, n may be
(a)5
(b)7
(c)9
(d)11
Q2.Find two whole numbers which, when multiplied together give an answer of 41.
Q3. What is the last digit of the number you get by multiplying the first 2002 odd prime numbers together?
a. 1
b. 3
c. 5
d. 7
e. 9
Q4. 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?
(1) 1050
(2) 540
(3) 1440
(4) 1590
Q5.A tennis tournament is held in a school where every student plays 1 game against every other student. There are 210 boy vs. boy games held and 300 girl vs. girl games. Q19. How many boy vs. girl games were held?
a. 40
b. 225
c. 525
d. 810
Q6. For how many values of k is 1212 the LCM of the positive integers 66, 88 and k?
(a) 24
(b) 1
(c) 25
(d) 25 x 13
Q7. A very fast growing sun-flower grows to a height of 12 feet in 12 weeks by doubling its height every week. If you only want your sun-flower to be 6 feet tall, after how many weeks should you stop it growing?
Q8. The sum of three consecutive numbers in a geometric sequence is 70. If the first is multiplied by 3, the second by 4 and the third by 4, the resulting numbers will be consecutive numbers in an arithmetic sequence. If each of the original numbers is a whole number, the first number is
a. 30
b. 70
c. 15
d. 25
e. 40
Q9. 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?
(1) (x-z)2 y is even
(2) (x-z)y2 is odd
(3) (x-z)y is odd
(4) (x-y)2z is even
Q10. Find the natural number below 1000, which has the maximum number of divisors. Find the sum of its digits?
a. 9
b. 18
c. 12
d. 15
Q11. (a + b + c + d + ...)23 = a23 + b23 + c23 + d23 + ....+ M, where M is divisible by
(a) 23
(b) 17
(c) 11
(d) can't be determined
Q12. Place an ordinary mathematical symbol between 2 and 3 so that the result is a number which greater than 2 and less than 3.
END
QBM047
Q1. The number FIVE as written using block capitals contains exactly 10 strokes or segments of a straight line. Find a number which when written out in words (using no tricks) contains as many strokes as the number says.
Q2. The number of 1's in the binary notation of 289 - 1 is
(a) 89
(b) 88
(c) 90
(d) 1
Q3. The highest power of 2 in 10! + 11! + 12! + 13! + ...+ 1000! is
(a) 8
(b) 9
(c) 10
(d) 11
Q4. How many natural numbers between 1 and 900 are NOT multiples of any of the numbers 2, 3, or 5?
a. 650
b. 660
c. 240
d. 250
Q5. Find the remainder when 51138 is divided by 7.
a. 2
b. 1
c. 2138
d. 3
ANS b
Q6. Assume that all bricklayers work at the same rate of speed. If it takes nine bricklayers (all working at the same time) fourteen days to do a job, how long would it take for the job to be done by
a) seven bricklayers
b) three bricklayers.
Q7. If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of a?
1. 121
2. 3267
3. 363
4. 33
Correct choice (3). Correct Answer - (363
Q8. What is the remainder when 62002 is divided by 11?
Q9. Let x = 1! + 2! + 3! + 4! + ... + 100!. Unit's digit of Xxxx...x is
a. 3
b. 1
c. 1 or 3 depending upon the number of times x appears in the power.
d. can't be determined.
ANS: d
Q10. Let (x, y) be co-prime numbers. Then
(a) x + y and x - y have no common factor other than 1 and 3 for all values of x and y.
(b) x + y and x - y have no common factor other than 1 and 2 for all values of x and y.
(c) x + y and x - y have no common factor other than 1 for all values of x and y.
(d) none of the above
Q11. What is the least number that should be multiplied to 100! To make it perfectly divisible by 718
a. 1
b. 7
c. 21
d. 49
Q12. A car has traveled 24,000 km and, in that distance, has worn out 6 tyres. Each tyre travelled the same distance. How far did each separate tyre travel?
Q13. There were 90 questions in an exam. If 1 mark was awarded for every correct answer and 1/3rd mark was deducted for every wrong answer, how many different net scores were possible in the exam?
a. 120
b. 359
c. 358
d. 360
ans: 358
Q14. If x = 15 x 30 x 45 .... 1500, then how many zeros are there at the end of x?
a. 24
b. 124
c. 97
d. 50
END