Number System (Q) Anand [PDF]

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Ajai Singh (Maths Legend)



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LEVEL # 1 How many numbers between 200 and 400 are divisible by 3, 5, 6? (a) 7 (b) 8 (c) 10 (d) 6 The largest natural number by which the product of three consecutive even natural numbers is always divisible is : (a) 16 (b) 24 (c) 48 (d) 96 The unit digit in the product (281 × 15 × 16 × 18) is : (a) 6 (b) 5 (c) 8 (d) 0 If n = 1 + x, where x is the product of 4 consecutive numbers then n is: (i) odd number (ii) even number (iii) prime (iv) square (a) i and iii (b) ii and iv (c) i and iv (d) None of these When any 4 digit number is multiplied by x, the 4-digit number repeats itself to form an 8-digit number. Then x is necessarily a multiple of : (a) 73 (b) 137 (c) both (d) None of these Yakov Perelman, the soviet mathematician died in 1871, he used to boast that he was x years old in the year x2. When was he born? (a) 1817 (b) 1849 (c) 1843 (d) 1806 Among how many students, 175 apple and 105 oranges can be equally divided? (a) 70 (b) 35 (c) 34 (d) 15



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A number lies between the cubes of 15 and 16. If the number is divisible by the square of 12 as well as by 7. What is the number? (a) 5249 (b) 4045 (c) 4032 (d) 3469 How many primes are there between 100 and 200? (a) 23 (b) 21 (c) 20 (d) 22 How many primes are there between 10! + 2 and 10! + 10? (a) 4 (b) 3 (c) 2 (d) None of these If the product (46A24) × 278 is divisible by 8, then A can’t be : (a) 2 (b) 4 (c) 5 (d) can be any value Sum of two irrational number is : (a) An integer (b) Always an irrational number (c) Always a rational number (d) None of these There are 4 consecutive even numbers; the product of 1st and 4th is 216. What will be the nd rd product of 2 and 3 . (a) 216 (b) 220 (c) 224 (d) None of these The difference between the squares of any two consecutive odd integers is always divisible by: (a) 8 (b) 9 (c) 11 (d) 6 Find a number, which when multiplied by 5 will exceed 60 as much as it is now short of 60 : (a) 25 (b) 15 (c) 20 (d) 10



Ajai Singh (Maths Legend)



16. If sum of three consecutive numbers is 18, then their product is : (a) 360 (b) 240 (c) 210 (d) 160 3 17. If x is odd, which of the following must be true 1. x is odd 2. x2 is odd 3. x2 is even (a) 1 only (b) 2 only (c) 1 and 3 only (d) None of these 18. The number 10n – 1 is divisible by 11 for : (a) All values of n (b) Odd values of n (c) Even values of n (d) None of these 19. Find the sum of numbers between 300 and 400 such that when they are divided by 6, 9 and 12 it leaves remainder as 4 in each case : (a) 1092 (b) 692 (c) 884 (d) 776 20. The least multiple of 7 which leaves a remainder of 4 when divided by 6, 9, 15 and 18 is : (a) 94 (b) 184 (c) 364 (d) 74 21. If acb + bca + bac + cab + cba = 3194, where abc is a three-digit number. Find the value of abc. (a) 338 (b) 358 (c) 368 (d) None of these 22. The product N of three positive integers is 6 times their sum. One of the integers is the sum of the other two. Find the sum of all possible values of N. (a) 316 (b) 326 (c) 346 (d) 336 23. The number from 1 to 33 are written side by side as follows : 2



123456 ...... 33. If this number is divided by 9, what will be the remainder? (a) 0 (b) 1 (c) 3 (d) 6 24. A 101 digit number 222...2X3...333 is formed by repeating the digit 2,50 times followed by the digit X and then repeating the digit 3, 50 times. The number is a multiple of 7, find the value of X? (a) 2 (b) 3 (c) 4 (d) 5 25. After div ision of a number by 3, 4, 7 the remainders are 2, 1, 4 respectiv ely. What will be the remainder if 84 divides the same number? (a) 80 (b) 53 (c) 79 (d) 81 26. Digital sum of a number is the sum of its digits. For example, the digital sum of 2003 = 5. The number of 4 digit numbers less than 2003 that have a digital sum less than that of 2003 is : (a) 20 (b) 21 (c) 22 (d) 23 61 62 63 64 27. 4 + 4 + 4 + 4 is divisible by: (a) 12 (b) 10 (c) 11 (d) 15 28. How many pairs of decimal digits are there with the property that if p is an integer ending in these two digits, then p2 ends in same two digits (repetition allowed) (a) 3 (b) 4 (c) 5 (d) 2 29. How many times the keys of a computer keyword need to be pressed while writing first 250 numbers i.e. from 1 to 250? (a) 632 (b) 642 (c) 645 (d) 640



Ajai Singh (Maths Legend)



30. What will be the unit digit of (463)4448? (a) 9 (b) 3 (c) 7 (d) 1 31. How many numbers between 500 and 1000 are divisible by 5 and 7 both? (a) 14 (b) 15 (c) 17 (d) 18 32. If the number 103AB gives 1 as a remainder when divided by 5, 7 and 11 in each case, then A + B is: (a) 12 (b) 14 (c) 13 (d) 15 33. How many 3-digit positive integers are perfect squares? (a) 20 (b) 23 (c) 22 (d) 21 34. Mr. Bansal, accountant of TCM subtracted 4 from a number and then divided the result by 5. Next, he subtracted 5 from the original number and then div ided the result by 5. Next, he subtracted 5 from the original number and then divided the result by 4. He got same final answer both times. The original number was : (a) 15 (b) 20 (c) 9 (d) 19 35. How many 2-digit prime numbers have the product of digits less than 10 : (a) 9 (b) 10 (c) 8 (d) 11



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A number N is divisible by 6 but not divisible by 4. Which of the following will not be an integer? 9. (a) N/3 (b) N/2 (c) N/6 (d) N/12 2 If n is a perfect cube, then which of the following statement is always true? 3



(a) n is odd (b) n is even 3 (c ) n is perfect square (d) n is a perfect cube For what values of x is 25 x + 1 divisible by 13? (a) All real values of x (b) Odd natural values of x (c) Even value of x (d) All the integral values of x What is the unit’s digit of 213 × 212 × 347 × 468 × 778? (a) 4 (b) 8 (c) 6 (d) 2 If the unit’s digit in the product (47n × 729 × 345 × 343) is 5, what is the maximum numbers of values that n may take? (a) 9 (b) 3 (c) 7 (d) 5 If a number is divided by 15, it leaves a remainder of 7. If thrice the number is divided by 5, then what is the remainder? (a) 5 (b) 6 (c) 7 (d) 1 A number when divided by 391 gives a remainder of 49. Find the remainder when it is divided by 39: (a) 10 (b) 9 (c) 11 (d) Cannot be determined (CND) Which of the following will divide 1112296 – 1? (a) 11 and 12 (b) 11 and 10 (c) 10 and 12 (d) 11 only If a, b, c and d are consecutive odd numbers, then (a2 + b2 + c2 + d2) is always divisible by : (a) 5 (b) 7 (c) 3 (d) 4



Ajai Singh (Maths Legend)



10. The last digit of the LCM of (32003 – 1) and (32003 + 1) is : (a) 8 (b) 2 (c) 4 (d) 6 11. The LCM of two numbers is 40 times their HCF. The sum of the LCM and HCF is 1.476. If one of the numbers is 288, find the other numbers? (a) 169 (b) 180 (c) 240 (d) 260 12. LCM of two natural numbers is 590 and their HCF is 59. How many sets of values are possible? (a) 1 (b) 2 (c) 5 (d) 10 13. How many zeroes will be there at the end of 25 × 35 × 40 × 50 × 60 × 65? (a) 6 (b) 8 (c) 5 (d) 7 14. What is the unit digit of 1! + 2! + 3! + .......+ 99! + 100!? (a) 3 (b) 1 (c) 5 (d) 6 3 15. N= 7! . How many factors of N are multiple of 10? (a) 736 (b) 1008 (c) 1352 (d) 894 16. How many zeroes will be there at the end of the expression N = 2 × 4× 6 × 8 × .... × 100? (a) 10 (b) 12 (c) 14 (d) None of these 17. How many zeroes will be there at the end of the expression N = 10 × 20 × 30 ..... × 1000? (a) 1280 (b) 1300 (c) 1320 (d) None of these



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18. How many zeroes will be there at the end of the expression N = 7 × 14 × 21 × .... × 777? (a) 24 (b) 25 (c) 26 (d) None of these 19. The number from 1 to 33 are written side by side as follows : 123456 ..... 33. What is the remainder when this number is divided by 9? (a) 0 (b) 1 (c) 3 (d) 6 20. The number 444444..... (999 times) is definitely divisible by : (a) 22 (b) 44 (c) 222 (d) All of these 21. Find the unit digit of N = 1727!37! (a) 1 (b) 3 (c) 7 (d) 9 22. How many zeroes will be there at the end of 1003 × 1001 × 999 × ... × 123? (a) 224 (b) 217 (c) 0 (d) None of these 23. How many zeroes will be there at the end of 361361? (a) 7 × 6! (b) 8 × 6! (c) 7 × 36! (d) 8 × 36! 2233



24. What is the unit digit of 711 ? (a) 1 (b) 3 (c) 7 (d) 9 150 76 140 25. A = 3 × 5 × 7 , B = 3148 × 576 × 7141, C = 3148 × 580 × 7139, D = 3151 × 580 × 7142, then the order of A, B, C and D from largest to smallest is : (a) DACB (b) CDBA (c) CDAB (d) DCAB



Ajai Singh (Maths Legend)



26. The first 20 natural numbers from 1 to 20 are written next to each other to form a 31 digit number N =123456789101121314151617181920. What is the remainder when this number is divided by 16? (a) 0 (b) 4 (c) 7 (d) 9 27. How many zeroes will be there at the end of N = 18! + 19!? (a) 3 (b) 4 (c) 5 (d) CND 28. Find the product of all the factors of 316 : (a) 333 (b) 368 (c) 3136 (d) 3128 29. What is the remainder when 9091 is divided by 13? (a) 0 (b) 7 (c) 12 (d) 1 30. Find the remainder when 90 91 is divided by 13? (a) 1 (b) 55 (c) 10 (d) 0 31. Find the remainder when 5555 2222 2222 + 5555 is divided by 7? (a) 1 (b) 3 (c) 0 (d) 5 32. What is the remainder when (103 + 93)752 is divided by 123? (a) 1 (b) 729 (c) 752 (d) 1000 33. How many divisors of 105 end with a zero? (a) 1 (b) 3 (c) 9 (d) 16 34. What is the remainder when 323232 is divided by 7? (a) 2 (b) 3 (c) 4 (d) 6 35. [111....111(200 digits) – 222... 22 (100 digits)]1/2 is equal to : (a) 1313 ..... 1313 (100 digits) (b) 2121 ..... 2121 (100 digits) 5



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(c) 1111 ...... 1111 (100 digits) (d) 3333 ...... 333 (100 digits) If p + p + 2, p + 4 are prime numbers, then the number of possible solution for p is : (a) 0 (b) 1 (c) 2 (d) None of these N! is having 37 zeroes at its end. How many values of N is/are possible? (a) 0 (b) 1 (c) 5 (d) Infinite P is a natural number. 2P has 28 divisors and 3P has 30 divisors. How many divisors of 6P will be there? (a) 35 (b) 40 (c) 45 (d) 48 What is the remainder when 323334 is divided by 7? (a) 4 (b) 2 (c) 3 (d) 1 Which of the following number is the greatest number dividing a family of numbers (N5 – N). Where N is any natural number? (a) 0 (b) 15 (c) 30 (d) 120 What is the remainder when [100! + 1] is divided by 101? (a) 100 (b) 1 (c) 0 (d) 10 Find the last non zero digit of 96!. (a) 2 (b) 4 (c) 6 (d) 8 What is the remainder when 2100 is divided by 101? (a) 1 (b) 100 (c) 99 (d) None of these How many divisors of 105 will have atleast one zero at its end? (a) 9 (b) 12 (c) 15 (d) 25 