Number System II [PDF]

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NUMBER SYSTEM - II



01. N is a positive integer such that when divided by 6, leaves a remainder of 5 and when divided by 8, leaves a remainder of 7. Which of the following is NOT a possible remainder, if N is divided by 120? A) 47



B) 23



C) 71



D) 59



E) 95



02. A number leaves a remainder of 18 when divided by 21. The number is doubled and then it is increased by 100. What remainder will this new number leave when divided by 7? A) 0



B) 2



C) 3



D) 4



E) 6



03. When the positive integer x is divided by 4, is the remainder equal to 3? x is divided by 2, the remainder is 1. 3 (2) x is divisible by 5.



(1) When



04. What is the remainder when 2n is divided by 10? (1) n is divisible by 6 (2) n is not divisible by 12 05. If the integer N lies between 100 and 150, N = ? (1) When N is divided by 7, the remainder is 5 (2) When N is divided by 6, the remainder is 1 06. If x is an integer greater than 1, is x even? (1)



x is not an integer. 4



(2) 2x leaves remainder 6 when divided by 20.



07. If each digit of the three-digit integer K is a positive multiple of 4, what is the value of K? (1) The unit’s digit of K is the least common multiple of the tens and hundreds digit of K. (2) K is NOT a multiple of 3. 08. What is the remainder when the two digit number PQ is divided by 3? (1) P × Q = 18. (2) Both P as well as Q are divisible by 5. 09. PQ denotes a two digit number. If R is a positive integer, what is the remainder when (PQ)R is divided by 5? (1) Q = 6. (2) R = 301.



10. 3q is a factor of



300! where q is a positive integer. What is the highest 100!



possible value of q? A) 67



B) 89



C) 100



D) 114



E) 148



11. 5n is a factor of 125! – 100! where n is a positive integer. What is the highest possible value of n? A) 5



B) 7



C) 18



D) 24



E) 31



12. How many numbers between 1 and 1,000 are divisible by 3 but not by 5, and by 7 but not by 9? A) 47



B) 26



C) 44



D) 23



E) 33



13. If abc represents a 3-digit positive integer, what is the remainder when 8abc is divided by 9? (1) c is odd. (2) abc = 125.



14. If p is a positive integer, for what minimum value of p is 7 10 p  4p divisible by 9? A) 7



B) 4



C) 3



D) 8



E) 5



15. If 48x is a factor of 200! where x is a positive integer, then the highest possible value of x is A) 197



B) 97



C) 95



D) 49



E) 33



16. For positive integers a and b, what is the remainder when 34a  b is divided by 5? (1) a = 2. (2) b = 5.



17. What is the highest power of 2 in 1! + 2! + 3! + … + 999! + 100!? A) 24



B) 3



C) 0



D) 97



E) NOTA



18. If a, b, c, d are positive integers greater than 1, which of the following can NEVER be a multiple of either one of a, b, c or d? A)



ab cd



B) ab + cd



C)



a c  b d



D)



a 2  b2 c2  d 2



E) abcd – 1



19. If N = 8,6M5 where M represents the digit at the ten’s place of N, what is the value of M? (1) N leaves a remainder of 1 when divided by 4 (2) N leaves a remainder of 5 when divided by 25



20. An integer is selected at random and raised to the fourth power. What is the probability that its last digit is greater than 1? A) 0.9



B) 0.6



C) 0.2



D) 0.5



E) 0.4



21. If p and r are positive integers, what is the remainder when p r is divided by 7? (1) p = 50 (2) r = 10



22. If p is a positive integer and q = p(p + 13), what is the remainder when q is divided by 8? (1) p leaves a remainder of 14 when divided by 16 (2) p leaves a remainder of 2 when divided by 4



23. If P is a positive integer, P = ? (1) When a positive integer N is divided by P, the remainder is 12. (2)



N  29.15 P



24. If P is a positive integer, what is the last digit of P508 ? (1) The last digit of P is a prime. (2) P is odd.



25. If P is a positive integer, what is the last digit of P604 ? (1) P is not a multiple of 5. (2) P is even.



26. A three digit integer N, when divided by 8 leaves remainder of 3, and when divided by 9 leaves remainder of 4. If 500 < N < 600, its last two digits are A) 31



B) 41



C) 51



D) 61



E) 71



27. If the unit’s digit of N is greater than 4, what is the unit’s digit of N? (1) The unit’s digit of N2 is not the same as that of N. (2) The unit’s digit of N3 is the same as that of N.



28. 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? A) 13



B) 59



C) 35



D) 37



E) 12



29. If y = 80x + 36, where x is a positive integer, which of the following can NEVER divide y completely? A) 3



B) 7



C) 12



D) 14



E) 24



30. When 58123 and 59059 are divided by a three digit number N, the



remainder obtained is same. How many values can N take? A) 3



B) 7



C) 5



D) 9



E) 8



31. On dividing 2272 as well as 875 by the 3-digit number N, we get the same remainder. What is the sum of the digits of N? A) 11



B) 10



C) 9



D) 8



E) 7



32. A number successively divided by 3, 4 and 7 leaves 2, 1 and 4 respectively as remainders. What will be the remainder if 84, divides the same number? A) 53



B) 80



C) 41



D) 76



33. What is the value of the integer n? (1) There are 31 zeros in n! (2) n is prime



E) 34



34. If N  13  23  33  ...  983  993 , what is the last digit of N? A) 0



B) 1



C) 9



D) 8



35. What would be the remainder if 1818 A) 1



B) 2



C) 3



36



E) 5



is divided by 7?



D) 4



E) 5



36. What would be the remainder if 43101  23101 is divided by 66? A) 1



B) 2



C) 10



D) 5



E) 0



37. Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The minimum number of boxes containing same number of oranges is A) 5



B) 3



C) 8



D) 6



38. What will be the remainder, when 1112 A) 8



B) 2



C) 1



E) 12 13



D) 5



is divided by 9? E) 7



39. When a positive integer N is divided by either 24 or 25, it leaves the same remainder. If N is divisible by 17, then the sum of digits of the least possible value of N is A) 11



B) 10



C) 9



D) 15



E) 7



40. Positive integers x and y leave remainders 4 and 7 when divided by 15 respectively. Which of the following is NOT a possible value of 3x – 2y? A) 463



B) 358



C) 163



D) 227



E) 913



41. The number N is such that (N – 6) is a multiple of 13. The largest number that will always divide (N + 7)(N + 20) is A) 140



B) 26



C) 169



D) 338



E) 150



42. F is the smallest natural number, which when multiplied by 7 gives a number made of 4’s only. Sum of the digits of F is G. The last digit of G92 is A) 4



B) 2



C) 9



D) 6



E) 8



43. Rose has a few identical chocolates; more than 80 but less than 100. She also has few identical bags. When she distributes 5 chocolates per bag, she is left with 4 chocolates for the last bag. When she distributes 6 chocolates per bag, she is left with 5 chocolates for the last bag. If she distributes 7 chocolates per bag, how many chocolates will be left for the last bag? A) 1



B) 5



C) 3



D) 6



E) 4



44. The integers p, q and r each leave the same remainder when divided by 9. What is the remainder when q is divided by 9? (1) p – r = q (2) p + r = q



45. The greatest number which on dividing 394 and 658, leaves remainders of 1 and 3 respectively is A) 3



B) 7



C) 9



D) 17



E) NOTA



46. The positive integers p, q and r each leave the same remainder when divided by 7. What is the value of q? (1) p – r = q (2) 30 < q < 40



47. What is the remainder if (3232 + 32) is divided by 33? A) 0



B) 32



C) 1



D) 2



E) 31



48. Two consecutive positive integers are divided by 9. What is the sum of the remainders obtained? (1) The sum of the remainders obtained is even. (2) The difference of the remainders obtained is 8.



49. What are the last two digits in the numerical value of 7507? A) 01



B) 07



C) 43



D) 49



E) 81



50. What is the remainder when a positive integer P is divided by 3? (1) P is the square of an even integer (2) P is not a multiple of 3



51. What is the remainder when the positive integer N is divided by 16? (1) N is the product of three consecutive even integers (2) Both 40 and 24 are factors of N



52. What is the remainder when X is divided by 4? (1) X is an odd integer (2) X is the square of a prime number greater than 3



53. What is the remainder when X is divided by 8? (1) X leaves a remainder of 1 when divided by 4 (2) X is the square of a prime number greater than 3



54. What is the rightmost non-zero digit in the integer value of 12!? A) 1



B) 2



C) 4



D) 6



E) 8



55. What smallest positive integer must be added to 1,000 so that the resultant number leaves remainders 1, 2 and 4 when divided by 4, 5 and 7 respectively? A) 97



B) 107



C) 117



D) 127



E) 137



56. What is the sum of the digits of the highest prime factor of 200!? A) 17



B) 13



C) 21



D) 14



E) 19



57. What is the sum of digits of the smallest number that leaves the same remainder of 4 when divided by 10, 11 or 12? A) 17



B) 16



C) 14



D) 12



E) 18



58. When a positive integer N is divided by another positive integer P, the N remainder is 15. If  29.075 , then P = P A) 8



B) 16



C) 24



D) 40



E) 200



59. When a positive integer p is divided by 7, the remainder is 4. When the same number is divided by 11, the remainder is 8. What smallest positive number must be added to p so that the new number becomes a multiple of 77? A) 3



B) 4



C) 7



D) 8



60. What is the value of the integer n? (1) n! ends in 28 zeros (2) (n + 1)! Ends in 31 zeros



E) 12



ANSWER KEY



01. D 02. C 03. E 04. C 05. E 06. B 07. E 08. E 09. A 10. C 11. D 12. B 13. D 14. E 15. D 16. B 17. C 18. E 19. C 20. D 21. A 22. A 23. C



24. E 25. C 26. E 27. C 28. D 29. E 30. B 31. B 32. A 33. C 34. A 35. A 36. E 37. D 38. C 39. C 40. D 41. D 42. D 43. B 44. D 45. E 46. C 47. A 48. D



49. C 50. C 51. A 52. B 53. B 54. D 55. C 56. E 57. B 58. E 59. A 60. C