Number System Workshop: Attempt all these questions and discussion will be held in class.
7. Find the smallest integer n such that n is a multiple of 2, (n + 1) is a multiple of 3, (n + 2) is a multiple of 5 and (n + 3) is a multiple of 7.
1. 53 2. 106 3. 316 4. 158 5. 263
8. What is the remainder when 7 + 707 + 70707 + 7070707 + 707070707 + 70707070707 + 7070707070707 is divided by 6?
1. 0 2. 1 3. 3 4. 5 5. 2
9. Find the number of different sets of three distinct co-prime natural numbers such that the LCM of the three numbers is 1001.
1. 8 2. 7 3. 6 4. 5 5. 4
10. In the entire career of Bobby Fischer, the well known chess player, he never lost 5 continuous successive games and nor did he ever win 7 continuous successive games. If he played a total of 1680 games in his career, find the difference between the maximum and the minimum number of games that he could have won.
1. 1104 2. 336 3. 1440 4. 1224 5. None of these
11. How many two digit numbers are such that the number is equal to a multiple of the product of its digits?
1. None 2. Two 3. Three 4. Five 5. More than five.
12. A two digit number is multiplied with the sum of the digits and the result is denoted by n. The same two digit number is then reversed i.e. the position of the digits is interchanged and the resulting number is again multiplied with the sum of its digits. This product is denoted as m. If n + m = 275. Find the sum of the digits of the original two digit number.
1. 3 2. 5 3. 7 4. 9 5. None of these
18. How many two digit natural numbers, n, exists such that 3n2 – 4n + 7 is divisible by 7?
1. None 2. 12 3. 13 4. 24 5. 26
19. How many distinct remainders are possible when 3n2 – 4n + 7, where n is a natural number, is divided by 5?
1. 1 2. 2 3. 3 4. 4 5. 5
20. What is the third non-zero digit from the right end in the product of the first 25 multiples of 5?
1. 3 2. 5 3. 6 4. 7 5. 8
26. What is the minimum number of natural numbers that should be selected from the first 100 natural numbers such that one is sure that four of the numbers selected add up to 202?
1. 2 2. 26 3. 51 4. 52 5. None of these
27. What is the maximum number of natural numbers that one can select from the first 99 natural numbers and be sure that no three of the selected numbers add up to 99?
1. 33 2. 67 3. 49 4. 66 5. None of these
28. The minimum number of equal squares that can be cut from a rectangular cardboard is 45. If both the length and breadth of the rectangle is a natural number between 18 and 40, find the area of the rectangle.
1. 315 2. 405 3. 540 4. 720 5. No unique value
29. A cuboid having dimensions 60 cm × 72 cm × 300 cm is cut into smaller cubes such that all the cubes cut are of the same size and the side of the cube is an integral measure. If n is the number of cubes so cut, how many different values could n assume?
1. 12 2. 6 3. 4 4. 3 5. 2
30. The product of two numbers, each greater than 1, is 175. Which of the following could be the LCM of the numbers?
1. 175 2. 35 3. 25 4. 25 or 35 5. 35 or 175
31. The distinct natural numbers, 30, n and 750 are such that the HCF of them is 30 and the LCM of them is 750. Which of the following could be a possible value of n?
1. 60 2. 90 3. 120 4. 150 5. No unique value
32. Find the number of ways to writing 64 as a product of three natural numbers.
1. 7 2. 8 3. 9 4. 10 5. None of these
45. How many natural numbers of the type ‘aabb’ are perfect squares?
1. None 2. One 3. Two 4. Three 5. More than three
46. How many two digit numbers exists that are completely divisible by the sum of their digits?
1. 8 2. 10 3. 15 4. 18 5. None of these
47. In how many ways can 60 be written as a sum of two natural numbers, both of which are neither a multiple of 2, nor a multiple of 3.
1. 9 2. 10 3. 11 4. 12 5. 13
48. What is the maximum number of natural numbers can be selected from the first 60 natural numbers such that the difference between no two of the numbers selected is equal to 12?
1. 48 2. 36 3. 24 4. 12 5. None of these
49. What is the maximum number of natural numbers that can be selected from 100 to 200, both inclusive, such that no two of the numbers selected add up to a multiple of 5?
1. 20 2. 21 3. 40 4. 41 5. 43
50. What is the maximum number of factors that a two digit natural number can have?
1. 7 2. 8 3. 12 4. 16 5. None of these
Directions for 51 & 52:
Three thieves and their pet monkey robbed certain number of coconuts. They slept at night, thinking that they would divide the coconuts in the morning. At night, first thief wakes up, divides the coconuts into three equal heaps, with one left-over, which he gave to the monkey. He takes one pile, mixes the remaining two piles and goes to sleep. Next, second thief wakes, divides the leftover coconuts into three equal heaps, with one left-over, which he gave to the monkey. He keeps one pile for himself and mixes the rest. After he slept, the same operation was done by the third thief. In the morning they divided the remaining coconuts in three equal heaps and one coconut was left, which was given to the monkey. Each took one heap for themselves.
51. What was the minimum number of coconuts they must have stolen?
1. 70 2. 25 3. 61 4. 79 5. None of these
52. If they stole the minimum number of coconuts, what is the number of coconuts that the second thief received (including the lot that he kept for himself at night)?
1. 33 2. 24 3. 18 4. 17 5. 11
53. How many natural numbers less than 1000 are such that when they are divided by 16, the result is a perfect fifth power and when they are multiplied by 8, the result is a perfect fourth power?
1. None 2. One 3. Two 4. Three 5. More than three
54. How many pairs of distinct natural numbers, a and b (a > b, both of them single digit numbers), do NOT satisfy the inequality ab < ba?
1. 2 2. 4 3. 6 4. 8 5. 10
55. Which of the following is the greatest? 248, 330, 425, 618, 1512
1. 248 2. 330 3. 425 4. 618 5. 1512
56. A grandfather divided the number of gold coins that he had, equally among his five sons. Two coins were left over which he kept for himself. His eldest son, divided his entire share among his four children and kept the two coins that were leftover with himself. The youngest son also divided his share among his three children and kept the two coins that were leftover for himself. Which of the following could be the number of coins the grandfather had?
1. 46 2. 56 3. 26 4. 131 5. 72
57. What is the highest power of 10 that divides 1001100 – 1?
1. 10 2. 100 3. 1000 4. 1001 5. None of these
66. If the unit digit of x99 is same as the unit digit of x97, what is the remainder when x2 is divided by 5? (x is a natural number not divisible 5)
1. 1 2. 2 3. 3 4. 4 5. No unique answer
67. 996 is a 12 digit number which reads as 9x1x801x9x01. If all the digits represented by x is the same digit, what digit is it?
1. 2 2. 3 3. 4 4. 5 6. 6
68. How many two digit numbers have exactly 12 factors?
1. 4 2. 5 3. 6 4. 7 5. 8
77. Two positive integers x and y such that x2 + 12y2 – 7xy – 4 = 0. How many values can y assume?
1. 2 2. 1 3. 3 4. 4
78. When a number N is divided by D, the remainder is 23 and when N is divided by 12D, the remainder is 104. What is the remainder when N is divided by 6D?
1. 47 2. 104 3. 52 4. 51
79. Let N = 233 × 321. How many positive divisors of N2 are less than N but do not divide N?
1. 1440 2. 747 3. 693 4.720
80. Find the highest power of 2 in the product of 1005 × 1006 × 1007 × …………………… × 2009.
1. 993 2. 995 3. 997 4. 1004