1 Number System 4. 2011 (II) 1. Consider the following statements I. The product of any three consecutive integers is divisible by 6. II. Any integer can be expressed in one of the three forms 3 k, 3 k + 1, 3 k + 2, where k is an integer. (a) 31 (b) II Only (d) Neither I nor II Explanation (c) I. The product of any three consecutive integers is 5. The largest integer that divides product of any four consecutive integers is (a) 4 (b) 6 (c) 12 (d) 24 Explanation (d) The largest integer that divides product of any four consecutive integers is 4! i.e., 24. e.g., 1, 2, 3, 4 are four consecutive integers. 6. 3k + 1 = {¼- 5, - 2, 1, 4, 7,¼ } and { 3k, 3k + 1, 3k + 2} = {¼ - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, 6, ¼ } always a natural number always a rational number always an irrational number either a natural number or an irrational number Explanation (d) If n is a natural number, then n is either a natural Hence, it is true. Which one of the following is a prime number? (b) 171 If n is a natural number, then n is (a) (b) (c) (d) 3k + 2 = {¼- 4, - 1, 2, 5, 8,¼ } (a) 161 (d) 53 Multiplication = 1 ´ 2 ´ 3 ´ 4 = 24 which divided by 24. II. Here, 3k = {¼- 6, - 3, 0, 3, 6,¼ } 2. (c) 49 always 1. Then, sum of numbers (2 + 3 + 5) = 10 is also divided on that number it will given a remainder 1. divisible by 3! i.e., 6. \ (b) 47 Explanation (a) If any number is divided by 2, 3 or 5 and remainder is Which of the above statements is/are correct? (a) I Only (c) Both I and II A number when divided by 2, 3 or 5 gives remainder 1. The number is (c) 173 (d) 221 number or an irrational number. When n = 3 Þ e. g ., When n = 9 Þ 3 = irrational number 9 = 3 = rational number Explanation (c) (a) Since, 161 < (13)2 If 161 is a prime number, then this is not divisible by any of the numbers, 2, 3, 5, 7, 11. But 161 is divisible by 7. Hence, 161 is not a prime number. (b) 171 < (14)2 For prime number, 171, is not divisible by any of the numbers 2, 3, 5, 7, 11, 13. But it is divisible by 3. Hence, 171 is not a prime number. (c) 173 < (14)2 For prime number 173, is not divisible by any of the numbers 2, 3, 5, 7, 11, 13. Hence, 173 is a prime number. (d) 221 < (15)2 For prime number 221, is not divisible by any of the numbers 2, 3, 5, 7, 11, 13. But it is divisible by 13. Hence, 221 is not a prime number. 3. Which among the following is the largest four digit number that is divisible by 88? (a) 9988 (c) 9944 (b) 9966 (d) 8888 Explanation (c) A number divisible by 88, it should be divisible by 8 and 11. In a given option, number 9944 and 8888 is divisible by 88. Hence, maximum number is 9944. 2011 (I) 7. ABC is a triangle and AD is perpendicular to BC. It is given that the lengths of AB, BC, CA are all rational numbers. Which one of the following is correct? (a) (b) (c) (d) AD and BD must be rational AD must be rational but BD need not be rational BD must be rational but AD need not be rational Neither AD nor BD need be rational Explanation (c) Since, D is a point of BC. As BC is rational so BD must be rational but AD need not be rational. 8. What is the number of prime factors of 30030? (a) Four (c) Six Explanation (c) (b) Five (d) None of these 2 3 5 11 7 13 30030 15015 5005 1001 91 13 1 4 CDS Chapterwise-Sectionwise Solved Papers Prime factors of 30030 = 2, 3, 5, 11, 7, 13 If 203 is a prime number then it is not divisible by any of the numbers 2, 3, 4, 5, 7, 11, 13,. But 203 is divisible by 7. Hence, 203 is not a prime number. \ Number of prime factors of 30030 is Six 9. If three sides of a right angled triangle are integers in their lowest form, then one of its sides is always divisible by (a) 6 (c) 7 (b) 5 (d) None of these Explanation (b) Let the lowest sides of a right triangle be 3, 4, 5. 2 2 14. The number 2784936 is divisible by which one of the following numbers? (a) 86 (c) 88 (b) 87 (d) 89 Explanation (c) Given number is 2784936. Sum of odd digits = 25 2 By pythagoras theorem ( 3) + ( 4) = ( 5) and Hence, one of its sides is always divisible by 5. If p is a prime such that p + 2 is also a prime, then I. p (p + 2) + 1 is a perfect square. II. 12 is a divisor of p + (p + 2), if p > 3. Which of the above statements is/are correct? (b) Only II (d) Neither I nor II II. p + 2 = 13 (prime number) 11 ´ 13 + 1 = 144 (a square number) Hence, both statements I and II are correct. 11. When a positive integer n is divided by 5, the remainder is 2. What is the remainder when the number 3n is divided by 5? (b) 2 (d) 4 Explanation (a) Let n = 5q + 2 (a) 1 (c) 4 [Q positive integer divisor of 12 = 1, 2, 3, 4, 6, 12 ] [Q positive integer divisor of 6 = 1, 2, 3, 6] = d( 4) [Q positive integer divisor of 4 = 1, 2, 4] =3 2010 (II) 16. The product of a rational number and an irrational number is (a) a natural number (c) a composite number (b) an irrational number (d) a rational number Explanation (b) We know that the product of a rational number and an term is 14, then what is the 95th term? = 5 ( 3 q + 1) + 1 When 3n is divided by 5, then remainder is 1. 12. Which one of the following three-digit numbers divides 9238 and 7091 with the same remainder in each case? (b) 209 (d) 191 Explanation (a) When we divide the number 9238 and 7091 by 113, we (a) - 75 (c) 80 13. Consider the following numbers (b) 75 (d) - 80 Explanation (a) Q T14 = 6 Þ and Þ …(i) a + 13 d = 6 T6 = 14 …(ii) a + 5 d = 14 On solving Eqs. (i) and (ii), we get a = 19, d = - 1 get the same remainder 85. \ T95 = a + 94 d = 19 - 94 = - 75 18. If n is a positive integer, then what is the digit in I. 247 II. 203 the unit place of 3 2 n + 1 + 2 2 n + 1 ? Which of the above numbers is/are prime? (a) Only I (c) Both I and II (b) 2 (d) None of these 17. If the 14th term of an arithmetic series is 6 and 6th 3n = 15 q + 6 (a) 113 (c) 317 of positive divisors of n. What is the value of d(d(d(12)))? irrational number is an irrational number. 3n = 3 ( 5 q + 2 ) Þ 15. For a positive integer n, define d (n) = The number (12 is a divisor of 24) 11 + 13 = 24 (a) 1 (c) 3 Hence, 2784936 is divisible by 88. Explanation (d) d (d (d (12 ))) = d (d ( 6)) Explanation (c) Taking p = 11 I. Difference = 25 - 14 = 11 \ 10. Consider the following statements (a) Only I (c) Both I and II sum of even digits = 14 (b) Only II (d) Neither I nor II Explanation (d) I Since, 247 < (16)2 If 247 is a prime number then it is not divisible by any of the numbers 2, 3, 5,7, 11, 13. But 247 is divisible by 13. Hence, 247 is not a prime number. (a) 0 (c) 5 (b) 3 (d) 7 Explanation (c) 32n + 1 + 2 2n + 1 = 32n × 3 + 2 2n × 2 If n is even, then unit’s place digit in 3 2n × 3 + 2 2n × 2 is 5 Q unit digit of 3 2n =1 and unit digit of 2 2n = 2 Number System and if n is odd, then unit’s place digit in 2n 3 ×3 + 2 Q unit digit of 3 2n 2n Given number is ABCDE. Here, A + C + E - ( B + D ) = 0 or divisible by 11 × 2 is 5 Hence, both statements are true. =9 and unit digit of 2 2n = 4 24. A three-digit number is divisible by 11 and has its digit in the unit’s place equal to 1. The number is 297 more than the number obtained by reversing the digits. What is the number? 2 19. If k is any even positive integer, then (k + 2 k) is (a) divisible by 24 (b) divisible by 8 but may not be divisible by 24 (c) divisible by 4 but may not be divisible by 8 (d) divisible by 2 but may not be divisible by 4 (a) 121 (c) 561 Explanation (b) If k is any even positive integer, then k 2 + 2 k is 2 2 which is divisible by 4. 25. Which one of the following numbers is not a square of any natural numbers? 20. What is the last digit in the expansion of 3 4798 ? (a) 1 (b) 3 (c) 7 (d) 9 (a) 5041 (c) 1936 4 1199 = Last digit in the expansion of ( 3 ) ×3 number may be 2, 3, 7, 8. 2 Hence, the number 9852 is not a square of any natural number. = Last digit in the expansion of 3 2 = 9 21. What is the value of x for which x, x + 1, x + 3 are all prime numbers? (a) 0 (c) 2 (b) 9852 (d) 6241 Explanation (b) Any number is not a square, if the unit’s place digit of Explanation (d) Last digit in the expansion of 34798 (b) 1 (d) 101 26. If r and s are any real numbers such that 0 £ s £ 1 and r + s = 1, then what is the maximum value of the product? 3 4 1 (d) 4 (a) 1 Explanation (c) If x = 2, then x + 1 and x + 3 are all prime numbers. (c) 2010 (I) (b) 1 2 Explanation (d) Given, r + s = 1 1 2 1 1 1 rs = ´ = 2 2 4 can be said about the expansion of - 6 4 n , where n is a positive integer? For maximum product, r = s = Last digit is 4 Last digit is 8 Last digit is 2 Last two digits are zero \ 22. What (a) (b) (c) (d) Explanation (d) Taking option (d), Now, 154 + 297 = 451 is equal to the original number. Let k = 2 m, m Î N, then k + k × 2 = 4m + 4m = 4m ( m + 1) 2 (b) 231 (d) 451 The reverse digit of 451 is 154. divisible by 8 but may not be divisible by 24. 12 n 27. The remainder on dividing given integers a and b by 7 are respectively 5 and 4. What is the remainder when ab is divided by 7? Explanation (d) 212n - 64 n = (212 )n - ( 64 )n = ( 4096)n - (1296)n n-1 = ( 4096 - 1296) [( 4096) n- 2 + ( 4096) n-1 (1296) + ¼+ (1296) ] (a) 3 (c) 5 (b) 4 (d) 6 Explanation (d) Let a = 7 p + 5 = 2800 ( k ) Hence, last two digits are always be zero. 23. Consider 5 the following assumption and two statements Assumption A number ‘ABCDE’ is divisible by 11. Statement I E - D + C - B + A is divisible by 11. Statement II E - D + C - B + A = 0 Which one of the following is correct? (a) Only statement I can be drawn from the assumption (b) Only statement II can be drawn from the assumption (c) Both the statements can be drawn from the assumption (d) Neither of the statements can be drawn from the assumption Explanation (c) We know that, if the difference of the sum of odd digits and sum of even digits is either 0 or multiple of 11, then the number is divisible by 11. and b = 7q + 4 where p and q are natural numbers. ab = (7 p + 5) (7q + 4) \ ab = 49 pq + ( 4 p + 5 q ) 7 + 20 = 7 {7 pq + 4 p + 5 q } + 7 ´ 2 + 6 when ab is divided by 7, we get the remainder 6. 2009 (II) 28. When a polynomial is divided by a linear polynomial, then what is the remainder? (a) (b) (c) (d) Constant polynomial only Zero polynomial only Either constant or zero polynomial Linear polynomial 6 CDS Chapterwise-Sectionwise Solved Papers Explanation (c) When a polynomial is divided by a linear polynomial, then the remainder is either constant or zero polynomial. third term is greater than the first by 9 and the second term is greater than the fourth by 18. What is the first term? x ( ax + b ) ax 2 + bx + c e.g., 32. There are four numbers forming a GP in which the (a) 2 (c) - 2 _ ax 2 + bx c = constant (b) 3 (d) - 3 Explanation (b) Let the GP series be OR a, ar, ar 2, ar 3, ar 4 ,¼¼ x ( ax + b ) ax 2 + bx By given condition, T3 = T1 + 9 ar 2 = a + 9 Þ _ ax 2 + bx 0 = zero 2 and 29. What least value must be given to Ä so that the number 84705 Ä 2 is divisible by 9? (a) 0 (c) 2 T2 = T4 + 18 Þ ar = ar 3 + 18 ar (1 - r 2 ) = 18 Þ (b) 1 (d) 3 \ is divisible by 3. a ( r 2 - 1) - ar ( r 2 - 1) a ( 4 - 1) = 9 a=3 Þ If we Ä = 1, then 26 + 1 = 27 is divisible by 9. 30. If p is an integer, then every square integer is of the form 2p or ( 4 p - 1) 4p or ( 4 p - 1) 3p or ( 3p + 1) 4p or ( 4 p + 1) 2009 (I) 33. When a natural number n is divided by 4, the remainder is 3. What is the remainder when 2n is divided by 4? (a) 1 (c) 3 Explanation (c) In option (c) When we put p = 1, 2, 3, 4,¼,we get (b) 2 (d) 6 Explanation (b) Let number be n = 4q + 3, where q is an integer. 3 or 4, 6 or 7, 9 or 11, 12 or 13, 15 or 16,… 2 n = 8q + 6 \ Hence, we get all square integers. 2 n = 4 (2q + 1) + 2 31. The angles of a triangle are in AP and the greatest angle is double the least. What is the ratio of angles in the radian measure? (a) 2 : 3 : 4 (c) 3 : 3 : 6 (b) 1 : 2 : 3 (d) 4 : 5 : 7 (7 )754 º (7 4 )188 ´ 7 2 a + 2d = 2 a (given condition) a = 2d ...(i) a + a + d + a + 2d = 180° Þ Þ 3a + 3 d = 180° æaö 3a + 3 ç ÷ = 180° from è2 ø 9a = 360° a = 40°, d = 20° \ Ratio of angles = 40° : 60°: 80° = 2 :3:4 º (7 )2 = 49 Hence, last digit is 9. 35. Consider the following statements (Q sum of angles of triangle = 180°) Þ 34. What is the last digit in the expansion of (2457)754 ? (a) 3 (b) 7 (c) 8 (d) 9 on (7 )754 . a, a + d , a + 2d . Þ When 2 n is divided by 4, then remainder will be 2. Explanation (d) The last digit in the expansion of (2457 )754 is depend Explanation (a) Let angles of a triangle in AP are Þ 9 18 - r=2 Þ = 26 + Ä Also, = \ From Eq. (i), Here, sum of digits = 8 + 4 + 7 + 0 + 5 + 2 + Ä Also, …(ii) On dividing Eq. (i) by Eq. (ii) Explanation (b) The given number is divisible by 9, if sum of the digits (a) (b) (c) (d) …(i) a ( r - 1) = 9 Þ Eq ...(i) A number a1 a2 a3 a4 a5 is divisible by 9, if I. a1 + a2 + a3 + a4 + a5 is divisible by 9. II. a1 - a2 + a3 - a4 + a5 is divisible by 9. Which of the above statements is / are correct? (a) Only I (c) Both I and II (b) Only II (d) Neither I nor II Explanation (a) As we know that a number a1 a2 a3 a4 a5 is divisible by 9, if sum of the digits, i .e., a1 + a2 + a3 + a4 + a5 is divisible by 9. Hence, only statement (I) is true. Number System 36. The product of two alternate odd integers exceeds three times the smaller by 12. What is the larger number? (a) 3 (c) 7 (b) 5 (d) 9 Sum of odd digit places = 8 + 9 + 5 + 6 + 5 = 33 Sum of even digit places = 7 + 8 + 4 + * = 19 + * Explanation (c) Let the first odd number be x and the alternate odd Now, 33 - (19 + *) = 14 - *, it is divisible by 11, if value of * is 3. number is x + 4. By given condition x ( x + 4) = 3 x + 12 42. The set of integers is closed with respect to which x 2 + 4 x = 3 x + 12 Þ one of the following? Þ x 2 + x - 12 = 0 Þ ( x + 4) ( x - 3) = 0 Þ x = 3, (Q x ¹ - 4) Hence, larger number is x + 4 = 7 37. If we divide a positive integer by another positive integer, what is the resulting number? (a) (b) (c) (d) 7 It is always a natural number It is always an integer It is a rational number It is an irrational number Explanation (c) When we divide a positive integer by another positive integer, the resultant will be a rational number. i.e., In the form of p/q, where p and q are positive integers and q ¹ 0 38. What is the total number of three digit numbers with unit digit 7 and divisible by 11? (a) 6 (c) 8 (b) 7 (d) 9 Explanation (c) The total number of three digit numbers with unit digit 7 and divisible by 11 are 187, 297, 407, 517, 627, 737, 847, 957. \ Total numbers = 8 (a) (b) (c) (d) Addition only Multiplication only Both addition and multiplication Division Explanation (c) The set of integers is closed with respect to addition and multiplication. e. g ., Let Z = {...- 3, - 2,- 1, 0, 1, 2, 3...} 1 + 2 = 3 and - 2 - 1 = - 3 1 ´ 2 = 2 and 2 ´ 1 = 2 (for addition) (for multiplication) 43. Which one of the following statements is always correct? (a) The square of a prime number is prime (b) The sum of two square numbers is a square number (c) The number of digits in a square number is even (d) The product of two square numbers is a square number Explanation (d) It is always correct that the product of two square numbers in a square number. e. g ., 4 ´ 9 = 36 44. A number, when divided by 987, gives a remainder 39. What is the sum of positive integers less than 100 59. When the same number is divided by 21, what is the remainder? which leave a remainder 1 when divided by 3 and leave a remainder 2 when divided by 4? Explanation (c) Let the number is x, when divided by 987, gives a (a) 416 (c) 1250 (b) 620 (d) 1314 (a) 21 \ (d) 15 x = 987 ´ k + 59 (Q kEZ + ) x = 47 ´ 21 ´ k + 21 ´ 2 + 17 10, 22, 34, 46, 58, 70, 82, 94 \ Total sum = 10 + 22 + 34 + 46 + 58 + 70 + 82 + 94 = 416 (c) 17 remainder Explanation (a) Required numbers are of the form of 12q - 2 i .e., (b) 19 x = 21( 47 k + 2 ) + 17 \ The number is divisible by 21, gives remainder 17 2008 (II) 2008 (I) 40. What is the sum of all prime numbers between 100 45. Which of the following numbers is a prime? (a) 667 (b) 861 (c) 481 (d) 331 and 120? (a) 652 (c) 644 (b) 650 (d) 533 Explanation (d) The prime numbers between 100 and 120 are 101, Explanation (d) Since, 667 = 1 ´ 23 ´ 29 861 = 1 ´ 7 ´ 3 ´ 41 103, 107, 109, 113. 481 = 1 ´ 13 ´ 37 \ Sum = 101 + 103 + 107 + 109 + 113 = 533 41. What least value must be given to * so that the number 8798546*5 is divisible by 11? (a) 0 (c) 2 (b) 1 (d) 3 Explanation (d) Any number is divisible by 11, if difference between sum of odd digits and sum of even digits is zero or multiple of 11. Let N=8798546*5 and 331 = 1 ´ 331 It is clear, from above, that prime number is 331. 46. One dividing 4996 by a certain number, the quotient is 62 and the remainder is 36. What is the divisor? (a) 80 (c) 90 (b) 85 (d) 95 8 CDS Chapterwise-Sectionwise Solved Papers Explanation (a) Let the divisor be x and we know that 2007 (II) Dividend = Quotient ´ Division + Remainder 50. If the numbers q, q + 2, q + 6 are all prime, then Also, given that dividend = 4996 what can be the value of 3 q + 9? Quotient = 62 and remainder = 36 \ 4996 = 62 ´ x + 36 Þ 62 x = 4996 - 36 4960 x= = 80 62 Þ (a) 18 Only (c) 60 Only Explanation (d) By taking options one by one. (a) 3 q + 9 = 18 Þ q = 3 \ Numbers are 3, 5, 9, which are not all prime. 47. Assertion (A) Zero is a whole number. (b) 3 q + 9 = 42 Þ q = 11 Reason (R) Every integer is a whole number. (a) A and R are correct, and R is correct explanation of A. (b) A and R are correct, but R is not correct explanation of A. (c) A is correct but R is wrong. (d) A is wrong but R is correct. Explanation (c) Zero is a whole number and every integer is not a whole number, because negative integers is not whole numbers. Therefore, A is correct and R is wrong. 48. A three-digit number has digits h, t, u (from left to right) with h > u. If the digits are reversed and the number thus formed is subtracted from the original number, the unit’s digit in the resulting number is 4. What are the other two digits of the resulting number from left to right? (a) 5 and 9 (b) 9 and 5 (c) 5 and 4 (d) 4 and 5 Explanation (a) Q Original number = h ´ 100 + t ´ 10 + u Number obtained by reversing digits = u ´ 100 + t ´ 10 + h \ Required number = ( h ´ 100 + t ´ 10 + u ) - (u ´ 100 + t ´ 10 + h) = 99 ( h - u ) But the unit’s place digit in above number is 4, therefore ( h - u ) should be 6, then number is 594. Whose digits are 5, 9, 4 respectively. 49. Let p denote the product 2 × 3 × 5 ¼59 × 61 of all primes from 2 to 61. Consider the sequence p + n (2 £ n £ 59). What is the number of primes in this sequence? (n is a natural number) (a) 0 (c) 17 (b) 16 (d) 58 Explanation (a) Given, p = 2 × 3 × 5¼¼59 × 61 = ...........0 Also, 2 £ n £ 59 Now, we check the sequence p + n. Since, unit digit of p is zero. Therefore, for every even value of n, ( p + n) is always divisible. For odd value of n = 3, 5, ¼¼ , 59 Take \ (b) 42 Only (d) Both (b) and (c) n=3 p + n = p + 3 = (2 × 3 × 5¼¼59 × 61 + 3) = 3 (2 × 5¼¼59 × 61 + 1), which is divisible. \ Numbers are 11, 13, 17, which are all prime. (c) 3 q + 9 = 60 Þ q = 17 Numbers are 17, 19, 23 which are all primes. Hence, option (d) is correct. 51. Which one of the following is correct? The sum of two irrational numbers (a) (b) (c) (d) is always a natural of irrational may be rational or irrational is always a rational number is always an irrational number Explanation (b) Let two rational numbers be 3 and 2 . \( 3 + 2 ) is an irrational number Now, Let two rational numbers be 3 - 2 , 2 . (3 - 2 ) + \ 2 = 3 (rational) Hence, sum of two irrational numbers may be rational or irrational. 52. Which one of the following is correct? The number 222222 is (a) (b) (c) (d) divisible by 3 but not divisible by 7 divisible by 3 and 7 but not divisible by 11 divisible by 2 and 7 but not divisible by 11 divisible by 3, 7 and 11 Explanation (d) Given, number is 222222. Here, sum of digits = 2 + 2 + 2 + 2 + 2 + 2 = 12 which is divisible by 3. So, given number is divisible by 3. Now, sum of odd terms of digits - Sum of even terms of digits = 6 - 6 = 0, it is divisible by 11. Since, in a number a digit repeated six times, then this number is divisible by 7, 11 and 13. \ The given number is divisible by 3, 7 and 11. 53. In a division operation, the divisor is 5 times the quotient and twice the remainder. If the remainder is 15, then what is the dividend? (a) 175 (b) 185 (c) 195 Explanation (c) Dividend = D ´ Q + R Given, D = 5 Q and D = 2 R When R = 15, D = 2 ´ 15 = 30 D 30 Q= = =6 5 5 Similarly, for even values of n, p + n is divisible. \ Hence, it is clear that p + n is always divisible by any number. So, there is no prime number exist in this sequence. \ Dividend = 30 ´ 6 + 15 = 195 (d) 250 Number System 54. If x and y denote respectively, the area and the sum of the length of diagonals of a rectangle with 1 length 1 unit and breadth unit, then which one of 2 the following is correct? (a) (b) (c) (d) x and y are rational x is irrational and y is irrational x is irrational and y is rational x and y are both irrational Explanation (b) d2 Explanation (a) Here, (25)2 > 589 Then, prime numbers less than 25 are 2, 3, 5, 7, 11, 13, 17, 19, 23. Since, 589 is divisible by 19, then 589 is a composite number. 57. What least number should be subtracted from 26492518 so that the resulting number is divisible by 3 but not by 9? (a) 1 (c) 4 (b) 3 (d) 7 Explanation (c) Sum of all the digits in the number 26492518 1 unit D 9 C = 2 + 6 + 4 + 9 + 2 + 5 + 1+ 8 d1 = 37 1 unit 2 Rectangle Obviously 4 is the least number which should be subtracted from the number 26492518. So that the resulting number is divisible by 3 but not by 9. 58. A ten-digit number is divisible by 4 as well as by 5. A B 1 1 Area = x = 1 ´ = = Rational 2 2 d 1 = d 2 = 12 + (1/2 )2 = y = d1 + d 2 = \ = 2 5 = 2 5 2 5 5 + 2 2 5 (a) 0, 1, 2, 4 or 6 (c) 2, 3, 4, 6 or 8 (b) 1, 2, 4, 6 or 8 (d) 0, 2, 4, 6 or 8 Explanation (d) Since, a ten-digit number is divisible by 4 as well as by 5, then this number must be divisible by 20. We known that any number is divisible by 20, if last two digits is divisible by 20. It means unit place will be zero and Ten’s place may be 0, 2, 4, 6 or 8. 59. By adding x to 1254934, the resulting number = Irrational 55. Which one of the following is correct? an + b n is divisible by a - b. (a) (b) (c) (d) What could be the possible digit at the ten’s place in the given number? for all integral values of n when n is an even integer when n is an odd integer for no integral value of n Explanation (d) When n is odd, then one of the factor of a n + b n is ( a + b ). When n is even, then there is no factor of a n + b n. Hence, a n + b n is not divisible by ( a - b ) for any integral value of n. becomes divisible by 11, while adding y to 1254934 makes the resulting number divisible by 3. Which one of the following is the set of values for x and y? (a) (b) (c) (d) x = 1, y = 1 x = 1, y = - 1 x = - 1, y = 1 x = - 1, y = - 1 Explanation (b) Difference of sums of even and odd places digit of 1254934 = (1 + 5 + 9 + 4) - (2 + 4 + 3) = 19 - 9 = 10 2007 (I) 56. Which one of the following numbers is a composite numbers? (a) 589 (c) 569 (b) 571 (d) 563 This number will be divisible by 11 after adding x, if the value of x is 1. Also, the sum of digits of 1254934 = 1 + 2 + 5 + 4 + 9 + 3 + 4 = 28 1254934 will be divisible by 3, after adding y, if the value of y is - 1.

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