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ANTITATIVE APTIT DE FOR COMPETITIVE EXAMINATIONS FULLY SOLVED AS PER NEW EXAMINATION PATTERN An ideal book for: Bank PO, SBI-PO, IBPS, RBI Exams MBA, MAT, CMAT, GMAT, CAT, IIFT, IGNOU SSC Combined Preliminary Exam, Hotel Management Sub-Inspectors of Police, CBI, CPO Exams UPSC- CSAT, SCRA and oth...

ANTITATIVE APTIT DE FOR COMPETITIVE EXAMINATIONS FULLY SOLVED AS PER NEW EXAMINATION PATTERN An ideal book for: Bank PO, SBI-PO, IBPS, RBI Exams MBA, MAT, CMAT, GMAT, CAT, IIFT, IGNOU SSC Combined Preliminary Exam, Hotel Management Sub-Inspectors of Police, CBI, CPO Exams UPSC- CSAT, SCRA and other State Services Exams Railway Recruitment Board Exams Campus Recruitment Tests With Latest Questions and Solutions S.CHAND Dr. R.S. AGGARWAL  Quantitative Aptitude For Competitive Examinations (Fully Solved) As per New Examination Pattern Disclaimer : While the author of this book have made every effort to avoid any mistakes or omission and have used their skill, expertise and knowledge to the best of their capacity to provide accurate and updated information. S Chand does not give any representation or warranty with respect to the accuracy or completeness of the contents of this publication and are selling this publication on the condition and understanding that they shall not be made liable in any manner whatsoever. S Chand expressly disclaims all and any liability/responsibility to any person, whether a purchaser or reader of this publication or not, in respect of anything and everything forming part of the contents of this publication. S Chand shall not be responsible for any errors, omissions or damages arising out of the use of the information contained in this publication. Further, the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purely coincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any individual. Quantitative Aptitude For Competitive Examinations (Fully Solved) As per New Examination Pattern An ideal book for : u Bank PO, SBI-PO, IBPS, RBI Exams u MBA, MAT, CMAT, GMAT, CAT, IIFT, IGNOU u SSC Combined Preliminary Exams, Hotel Management u Sub-Inspectors of Police, CBI, CPO Exams u UPSC-CSAT, SCRA and other State Services Exams u Railway Recruitment Board Exams u Campus Recruitment Tests Dr. R.S. AGGARWAL S Chand And Company Limited (AN ISO 9001 : 2008 COMPANY) RAM NAGAR, NEW DELHI - 110 055  S Chand And Company Limited (An ISO 9001 : 2008 Company) S. CHAND Head Office: 7361, RAM NAGAR, NEW DELHI - 110 055 empowering minds Phone: 23672080-81-82, 9899107446, 9911310888 Fax: 91-11-23677446 P U B L I S H I N G www.schandpublishing.com; e-mail: [email protected] Branches : Ahmedabad : Ph: 27541965, 27542369, [email protected] Bengaluru : Ph: 22268048, 22354008, [email protected] Bhopal : Ph: 4274723, 4209587, [email protected] Chandigarh : Ph: 2625356, 2625546, [email protected] Chennai : Ph: 28410027, 28410058, [email protected] Coimbatore : Ph: 2323620, 4217136, [email protected] (Marketing Office) Cuttack : Ph: 2332580, 2332581, [email protected] Dehradun : Ph: 2711101, 2710861, [email protected] Guwahati : Ph: 2738811, 2735640, [email protected] Hyderabad : Ph: 27550194, 27550195, [email protected] Jaipur : Ph: 2219175, 2219176, [email protected] Jalandhar : Ph: 2401630, 5000630, [email protected] Kochi : Ph: 2378740, 2378207-08, [email protected] Kolkata : Ph: 22367459, 22373914, [email protected] Lucknow : Ph: 4026791, 4065646, [email protected] Mumbai : Ph: 22690881, 22610885, [email protected] Nagpur : Ph: 6451311, 2720523, 2777666, [email protected] Patna : Ph: 2300489, 2302100, [email protected] Pune : Ph: 64017298, [email protected] Raipur : Ph: 2443142, [email protected] (Marketing Office) Ranchi : Ph: 2361178, [email protected] Siliguri : Ph: 2520750, [email protected] (Marketing Office) Visakhapatnam : Ph: 2782609, [email protected] (Marketing Office) © 1989, Dr. R.S. Aggarwal All rights reserved. No part of this publication may be reproduced or copied in any material form (including photocopying or storing it in any medium in form of graphics, electronic or mechanical means and whether or not transient or incidental to some other use of this publication) without written permission of the publisher. Any breach of this will entail legal action and prosecution without further notice. Jurisdiction : All disputes with respect to this publication shall be subject to the jurisdiction of the Courts, Tribunals and Forums of New Delhi, India only. S. CHAND’S Seal of Trust In our endeavour to protect you against counterfeit/fake books, we have pasted a holographic film over the cover of this book. The hologram displays the unique 3D multi-level, multi-colour effects of our logo from different angles when tilted or properly illuminated under a single source of light, such as 2D/3D depth effect, kinetic effect, gradient effect, trailing effect, emboss effect, glitter effect, randomly sparkling tiny dots, etc. A fake hologram does not display all these effects. First Edition 1989 Subsequent Editions and Reprints 1995, 1996 (Twice), 97, 98, 99, 2000, 2001, 2002, 2003, 2004, 2005 (Twice), 2006 (Twice), 2007 (Twice), 2008 (Twice), 2009 (Thrice), 2010 (Twice), 2011 (Thrice), 2012, 2013 (Twice), 2014 (Twice), 2015 (Twice), 2016 (Twice) Revised and Enlarged Edition 2017; Reprint 2017 ISBN : 978-93-525-3402-9 PRINTED IN INDIA By Vikas Publishing House Pvt. Ltd., Plot 20/4, Site-IV, Industrial Area Sahibabad, Ghaziabad-201010 and Published by S Chand And Company Limited, 7361, Ram Nagar, New Delhi-110 055.  Preface to the Revised Edition Ever since its release in 1989, Quantitative Aptitude has come to acquire a special place of respect and acceptance among students and aspirants appearing for a wide gamut of competitive exams. As a front-runner and a first choice, the book has solidly stood by the students and helped them fulfil their dreams by providing a strong understanding of the subject and even more rigorous practice of it. Now, more than a quarter of a century later, with the ever changing environment of examinations, the book too reinvents itself while being resolute to its core concept of providing the best content with easily understandable solutions. Following are the features of this revised and enlarged edition: 1. Comprehensive: With more than 5500 questions (supported with answers and solutions—a hallmark of Quantitative Aptitude) the book is more comprehensive than ever before. 2. Easy to follow: Chapters begin with easy-to-grasp theory complemented by formulas and solved examples. They are followed by a wide-ranging number of questions for practice. 3. Latest: With questions (memory based) from examinations up till year 2016, the book captures the latest examination patterns as well as questions for practice. With the above enhancements to an already robust book, we fulfil a long-standing demand of the readers to bring out a revised and updated edition, and sincerely hope they benefit immensely from it. Constructive suggestions for improvement of this book will be highly appreciated and welcomed. All the Best!  Salient Features of the Book u A whole lot of objective-type questions, with their solutions by short-cut methods. u A full coverage of every topic via fully solved examples given at the beginning of each chapter. u A separate exercise on Data-Sufficiency-Type Questions given in each topic, along with explanatory solutions. u A more enriched section on Data Interpretation. u Questions from latest years’ examination papers (on memory basis) have been incorporated.  Contents SECTION–I : ARITHMETICAL ABILITY 21. Alligation or Mixture 633–640 1–883 22. Simple Interest 641–662 1. Number System 3–50 23. Compound Interest 663–687 2. H.C.F. and L.C.M. of Numbers 51–68 24. Area 688–765 3. Decimal Fractions 69–94 25. Volume and Surface Area 766–813 4. Simplification 95–179 26. Races and Games of Skill 814–818 5. Square Roots and Cube Roots 180–205 27. Calendar 819–822 6. Average 206–239 28. Clocks 823–833 7. Problems on Numbers 240–263 29. Stocks and Shares 834–840 8. Problems on Ages 264–277 30. Permutations and Combinations 841–849 9. Surds and Indices 278–296 31. Probability 850–860 10. Logarithms 297–307 32. True Discount 861–865 11. Percentage 308–373 33. Banker’s Discount 866–869 12. Profit and Loss 374–425 34. Heights and Distances 870–876 13. Ratio and Proportion 426–475 35. Odd Man Out and Series 877–883 14. Partnership 476–492 15. Chain Rule 493–509 SECTION–II : DATA INTERPRETATION 16. Pipes and Cisterns 510–525 885–952 17. Time and Work 526–561 36. Tabulation 887–904 18. Time and Distance 562–599 37. Bar Graphs 905–922 19. Boats and Streams 600–611 38. Pie Chart 923–936 20. Problems on Trains 612–632 39. Line Graphs 937–952 Section–I Arithmetical Ability  1 Number System FUNDAMENTAL CONCEPTS I. Numbers In Hindu-Arabic system, we have ten digits, namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. A number is denoted by a group of digits, called numeral. For denoting a numeral, we use the place-value chart, given below. Ten- Crores Ten- Lakhs Ten- Thousands Hundreds Tens Ones Crores Lakhs Thousands (i) 5 2 8 6 7 9 (ii) 4 3 8 0 9 6 7 (iii) 3 5 2 1 8 0 0 9 (iv) 5 6 1 3 0 7 0 9 0 The four numerals shown above may be written in words as: (i) Five lakh twenty-eight thousand six hundred seventy-nine (ii) Forty-three lakh eighty thousand nine hundred sixty-seven (iii) Three crore fifty-two lakh eighteen thousand nine (iv) Fifty-six crore thirteen lakh seven thousand ninety Now, suppose we are given the following four numerals in words: (i) Nine crore four lakh six thousand two (ii) Twelve crore seven lakh nine thousand two hundred seven (iii) Four lakh four thousand forty (iv) Twenty-one crore sixty lakh five thousand fourteen Then, using the place-value chart, these may be written in figures as under: Ten- Crores Ten- Lakhs Ten- Thousands Hundreds Tens Ones Crores Lakhs Thousands (i) 9 0 4 0 6 0 0 2 (ii) 1 2 0 7 0 9 2 0 7 (iii) 4 0 4 0 4 0 (iv) 2 1 6 0 0 5 0 1 4 II. Face value and Place value (or Local Value) of a Digit in a Numeral (i) The face value of a digit in a numeral is its own value, at whatever place it may be. Ex. In the numeral 6872, the face value of 2 is 2, the face value of 7 is 7, the face value of 8 is 8 and the face value of 6 is 6. (ii) In a given numeral: Place value of ones digit = (ones digit) × 1, Place value of tens digit = (tens digit) × 10, Place value of hundreds digit = (hundreds digit) × 100 and so on. Ex. In the numeral 70984, we have Place value of 4 = (4 × 1) = 4, Place value of 8 = (8 × 10) = 80, Place value of 9 = (9 × 100) = 900, Place value of 7 = (7 × 10000) = 70000. Note: Place value of 0 in a given numeral is 0, at whatever place it may be. III. Various Types of Numbers 1. Natural Numbers: Counting numbers are called natural numbers. Thus, 1, 2, 3, 4,........... are all natural numbers. 2. Whole Numbers: All counting numbers, together with 0, form the set of whole numbers. Thus, 0, 1, 2, 3, 4,........... are all whole numbers. 3 4 QUANTITATIVE APTITUDE 3. Integers: All counting numbers, zero and negatives of counting numbers, form the set of integers. Thus,..........., – 3, – 2, – 1, 0, 1, 2, 3,........... are all integers. Set of positive integers = {1, 2, 3, 4, 5, 6,...........} Set of negative integers = { – 1, – 2, – 3, – 4, – 5, – 6,...........} Set of all non-negative integers = {0, 1, 2, 3, 4, 5,...........} 4. Even Numbers: A counting number divisible by 2 is called an even number. Thus, 0, 2, 4, 6, 8, 10, 12,........... etc. are all even numbers. 5. Odd Numbers: A counting number not divisible by 2 is called an odd number. Thus, 1, 3, 5, 7, 9, 11, 13,........... etc. are all odd numbers. 6. Prime Numbers: A counting number is called a prime number if it has exactly two factors, namely itself and 1. Ex. All prime numbers less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 7. Composite Numbers: All counting numbers, which are not prime, are called composite numbers. A composite number has more than 2 factors. 8. Perfect Numbers: A number, the sum of whose factors (except the number itself), is equal to the number, is called a perfect number, e.g. 6, 28, 496 The factors of 6 are 1, 2, 3 and 6. And, 1 + 2 + 3 = 6. The factors of 28 are 1, 2, 4, 7, 14 and 28. And, 1 + 2 + 4 + 7 + 14 = 28. 9. Co-primes (or Relative Primes): Two numbers whose H.C.F. is 1 are called co-prime numbers, Ex. (2, 3), (8, 9) are pairs of co-primes. 10. Twin Primes: Two prime numbers whose difference is 2 are called twin-primes, Ex. (3, 5), (5, 7), (11, 13) are pairs of twin-primes. p 11. Rational Numbers: Numbers which can be expressed in the form , where p and q are integers and q ≠ 0, are q called rational numbers. 1 –8 2 Ex. , , 0, 6, 5 etc. 8 11 3 12. Irrational Numbers: Numbers which when expressed in decimal would be in non-terminating and non-repeating form, are called irrational numbers. Ex. 2 , 3 , 5 , 7 , π, e , 0.231764735...... IV. Important Facts: 1. All natural numbers are whole numbers. 2. All whole numbers are not natural numbers. 0 is a whole number which is not a natural number. 3. Even number + Even number = Even number Odd number + Odd number = Even number Even number + Odd number = Odd number Even number – Even number = Even number Odd number – Odd number = Even number Even number – Odd number = Odd number Odd number – Even number = Odd number Even number × Even number = Even number Odd number × Odd number = Odd number Even number × Odd number = Even number 4. The smallest prime number is 2. 5. The only even prime number is 2. 6. The first odd prime number is 3. 7. 1 is a unique number – neither prime nor composite. 8. The least composite number is 4. 9. The least odd composite number is 9. 10. Test for a Number to be Prime: Let p be a given number and let n be the smallest counting number such that n2 ≥ p. Now, test whether p is divisible by any of the prime numbers less than or equal to n. If yes, then p is not prime otherwise, p is prime. NUMBER SYSTEM 5 Ex. Test, which of the following are prime numbers? (i) 137 (ii) 173 (iii) 319 (iv) 437 (v) 811 Sol. (i) We know that (12)2 > 137. Prime numbers less than 12 are 2, 3, 5, 7, 11. Clearly, none of them divides 137. ∴ 137 is a prime number. (ii) We know that (14)2 > 173. Prime numbers less than 14 are 2, 3, 5, 7, 11, 13. Clearly, none of them divides 173. ∴ 173 is a prime number. (iii) We know that (18)2 > 319. Prime numbers less than 18 are 2, 3, 5, 7, 11, 13, 17. Out of these prime numbers, 11 divides 319 completely. ∴ 319 is not a prime number. (iv) We know that (21)2 > 437. Prime numbers less than 21 are 2, 3, 5, 7, 11, 13, 17, 19. Clearly, 437 is divisible by 19. ∴ 437 is not a prime number. (v) We know that (30)2 > 811. Prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Clearly, none of these numbers divides 811. ∴ 811 is a prime number. V. Important Formulae: (i) (a + b)2 = a2 + b2 + 2ab (ii) (a – b)2 = a2 + b2 – 2ab (iii) 2 2 2 (a + b) + (a – b) = 2(a + b ) 2 (iv) (a + b)2 – (a – b)2 = 4ab (v) 3 3 3 (a + b) = a + b + 3ab (a + b) (vi) (a – b)3 = a3 – b3 – 3ab (a – b) (vii) 2 2 a – b = (a + b) (a – b) (viii) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc +  ca) (ix) a3 + b3 = (a + b) (a2 + b2 – ab) (x) a3 – b3 = (a – b) (a2 + b2 + ab) (xi) 3 3 3 2 2 2 a + b + c – 3abc = (a + b + c) (a + b + c – ab – bc – ca) (xii) If a + b + c = 0, then a3 + b3 + c3 = 3abc TESTS OF DIVISIBILITY 1. Divisibility By 2: A number is divisible by 2 if its unit digit is any of 0, 2, 4, 6, 8. Ex. 58694 is divisible by 2, while 86945 is not divisible by 2. 2. Divisibility By 3: A number is divisible by 3 only when the sum of its digits is divisible by 3. Ex. (i) In the number 695421, the sum of digits = 27, which is divisible by 3. ∴ 695421 is divisible by 3. (ii) In the number 948653, the sum of digits = 35, which is not divisible by  3. ∴ 948653 is not divisible by 3. 3. Divisibility By 9: A number is divisible by 9 only when the sum of its digits is divisible by 9. Ex. (i) In the number 246591, the sum of digits = 27, which is divisible by 9. ∴ 246591 is divisible by 9. (ii) In the number 734519, the sum of digits = 29, which is not divisible by  9. ∴ 734519 is not divisible by 9. 4. Divisibility By 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Ex. (i) 6879376 is divisible by 4, since 76 is divisible by 4. (ii) 496138 is not divisible by 4, since 38 is not divisible by 4. 5. Divisibility By 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8. 6 QUANTITATIVE APTITUDE Ex. (i) In the number 16789352, the number formed by last 3 digits, namely 352 is divisible by 8. ∴ 16789352 is divisible by 8. (ii) In the number 576484, the number formed by last 3 digits, namely 484 is not divisible by 8. ∴ 576484 is not divisible by 8. 6. Divisibility By 10: A number is divisible by 10 only when its unit digit is 0. Ex. (i) 7849320 is divisible by 10, since its unit digit is 0. (ii) 678405 is not divisible by 10, since its unit digit is not 0. 7. Divisibility By 5: A number is divisible by 5 only when its unit digit is 0 or 5. Ex. (i) Each of the numbers 76895 and 68790 is divisible by 5. 8. Divisibility By 11: A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11. Ex. (i) Consider the number 29435417. (Sum of its digits at odd places) – (Sum of its digits at even places) = (7 + 4 + 3 + 9) – (1 + 5 + 4 + 2) = (23 – 12) = 11, which is divisible by   11. ∴ 29435417 is divisible by 11. (ii) Consider the number 57463822. (Sum of its digits at odd places) – (Sum of its digits at even places) = (2 + 8 + 6 + 7) – (2 + 3 + 4 + 5) = (23 – 14) = 9, which is not divisible by  11. ∴ 57463822 is not divisible by 11. 9. Divisibility By 25: A number is divisible by 25 if the number formed by its last two digits is either 00 or divisible by 25. Ex. (i) In the number 63875, the number formed by last 2 digits, namely 75 is divisible by 25. ∴ 63875 is divisible by 25. (ii) In the number 96445, the number formed by last 2 digits, namely 45 is not divisible by 25. ∴ 96445 is not divisible by 25. 10. Divisibility By 7 or 13: Divide the number into groups of 3 digits (starting from right) and find the difference between the sum of the numbers in odd and even places. If the difference is 0 or divisible by 7 or 13 (as the case may be), it is divisible by 7 or 13. Ex. (i) 4537792 → 4 / 537 / 792 (792 + 4) – 537 = 259, which is divisible by 7 but not by 13. ∴ 4537792 is divisible by 7 and not by 13. (ii) 579488 → 579 / 488 579 – 488 = 91, which is divisible by both 7 and 13. ∴ 579488 is divisible by both 7 and 13. 11. Divisibility By 16: A number is divisible by 16, if the number formed by its last 4 digits is divisible by 16. Ex. (i) In the number 463776, the number formed by last 4 digits, namely 3776, is divisible by 16. ∴ 463776 is divisible by 16. (ii) In the number 895684, the number formed by last 4 digits, namely 5684, is not divisible by 16. ∴ 895684 is not divisible by 16. 12. Divisibility By 6: A number is divisible by 6, if it is divisible by both 2 and 3. 13. Divisibility By 12: A number is divisible by 12, if it is divisible by both 3 and 4. 14. Divisibility By 15: A number is divisible by 15, if it is divisible by both 3 and 5. 15. Divisibility By 18: A number is divisible by 18, if it is divisible by both 2 and 9. 16. Divisibility By 14: A number is divisible by 14, if it is divisible by both 2 and 7. 17. Divisibility By 24: A given number is divisible by 24, if it is divisible by both 3 and  8. 18. Divisibility By 40: A given number is divisible by 40, if it is divisible by both 5 and  8. 19. Divisibility By 80: A given number is divisible by 80, if it is divisible by both 5 and  16. NUMBER SYSTEM 7 Note: If a number is divisible by p as well as q, where p and q are co-primes, then the given number is divisible by pq. If p and q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q. Ex. 36 is divisible by both 4 and 6, but it is not divisible by (4 × 6) = 24, since 4 and 6 are not co-primes. VI. Factorial of a Number Let n be a positive integer. Then, the continued product of first n natural numbers is called factorial n, denoted by n ! or n. Thus, n ! = n (n – 1) (n – 2)........... 3.2.1 Ex. 5 ! = 5 × 4 × 3 × 2 × 1 = 120. Note: 0 ! = 1 VII. Modulus of a Number  x , when x ≥ 0 |x| =   – x , when x < 0 Ex.|– 5| = 5, |4| = 4, |– 1| = 1, etc. VIII. Greatest Integral Value The greatest integral value of an integer x, denoted by [x], is defined as the greatest integer not exceeding x.  11   3  Ex. [1.35] = 1,   =  2  = 2, etc.  4   4 IX. Multiplication BY Short cut Methods 1. Multiplication By Distributive Law: (i) a × (b + c) = a × b + a × c (ii) a × (b – c) = a × b – a × c Ex. (i) 567958 × 99999 = 567958 × (100000 – 1) = 567958 × 100000 – 567958 × 1 = (56795800000) – 567958) = 56795232042. (ii) 978 × 184 + 978 × 816 = 978 × (184 + 816) = 978 × 1000 = 978000. 2. Multiplication of a Number By 5n: Put n zeros to the right of the multiplicand and divide the number so formed by 2n. 9754360000 Ex. 975436 × 625 = 975436 × 54 = = 609647500. 16 X. Division Algorithm or Euclidean Algorithm If we divide a given number by another number, then: Dividend = (Divisor × Quotient) + Remainder Important Facts: 1. (i) (xn – an) is divisible by (x – a) for all values of n. (ii) (xn – an) is divisible by (x + a) for all even values of n. (iii) (xn + an) is divisible by (x + a) for all odd values of n. 2. To find the highest power of a prime number p in n ! n  n   n  n r r +1 Highest power of p in n! =   +  2  +  3  +...... +  r  , where p ≤ n < p p p  p  p  SOLVED EXAMPLES Ex. 1. Simplify: (i) 8888 + 888 + 88 + 8 (ii) 715632 – 631104 – 9874 – 999 (LIC, ADO, 2007) Sol. (i) 8 8 8 8 (ii) Given exp = 715632 – (631104 + 9874 + 99) 888 = 715632 – 641077 = 74555. 88 631104 715632 + 8 9874 – 641077 9 8 7 2 + 99 74555 641077 8 QUANTITATIVE APTITUDE Ex. 2. What value will replace the question mark in each of the following questions? (i)? – 1936248 = 1635773 (ii) 9587 –? = 7429 – 4358 Sol. (i) Let x – 1936248 = 1635773. Then, x = 1635773 + 1936248 = 3572021. (ii) Let 9587 – x = 7429 – 4358. Then, 9587 – x = 3071 ⇒ x = 9587 – 3071 = 6516. 1 2 Ex. 3. What could be the maximum value of Q in the following equation? 5 P 9 5P9 + 3R7 + 2Q8 = 1114 3 R 7 Sol. We may analyse the given equation as shown: 2 Q 8 Clearly, 2 + P + R + Q = 11. 1 1 1 4 So, the maximum value of Q can be (11 – 2), i.e. 9 (when P = 0, R = 0). Ex. 4. Simplify: (i) 5793405 × 9999 (ii) 839478 × 625 Sol. (i) 5793405 × 9999 = 5793405 × (10000 – 1) = 57934050000 – 5793405 = 57928256595. 4  10  839478 × 10 4 8394780000 (ii) 839478 × 625 = 839478 × 54 = 839478 ×   = = = 524673750.  2  24 16 Ex. 5. Evaluate: (i) 986 × 137 + 986 × 863 (ii) 983 × 207 – 983 × 107 Sol. (i) 986 × 137 + 986 × 863 = 986 × (137 + 863) = 986 × 1000 = 986000. (ii) 983 × 207 – 983 × 107 = 983 × (207 – 107) = 983 × 100 = 98300. Ex. 6. Simplify: (i) 1605 × 1605 (ii) 1398 × 1398 Sol. (i) 1605 × 1605 = (1605)2 = (1600 + 5)2 = (1600)2 + 52 + 2 × 1600 × 5 = 2560000 + 25 + 16000 = 2576025. (ii) 1398 × 1398 = (1398)2 = (1400 – 2)2 = (1400)2 + 22 – 2 × 1400 × 2 = 1960000 + 4 – 5600 = 1954404. Ex. 7. Evaluate: (i) 475 × 475 + 125 × 125 (ii) 796 × 796 – 204 × 204 1 2 2 Sol. (i) We have (a2 + b2) = [( a + b) + ( a – b) ] 2 1 1 \ (475)2 + (125)2 =. [(475 + 125)2 + (475 – 125)2 ] =. [(600)2 + (350)2 ] 2 2 1 1 = [360000 + 122500] = × 482500 = 241250. 2 2 (ii) 796 × 796 – 204 × 204 = (796)2 – (204)2 = (796 + 204) (796 – 204) [ (a – b)2 = (a + b)(a – b)] = (1000 × 592) = 592000. Ex. 8. Simplify: (i) (387 × 387 + 113 × 113 + 2 × 387 × 113) (ii) (87 × 87 + 61 × 61 – 2 × 87 × 61) Sol. (i) Given Exp. = (387)2 + (113)2 + 2 × 387 × 113 = (a2 + b2 + 2ab), where a = 387 and b = 113 = (a + b)2 = (387 + 113)2 = (500)2 = 250000. (ii) Given Exp. = (87)2 + (61)2 – 2 × 87 × 61 = (a2 + b2 – 2ab), where a = 87 and b = 61 = (a – b)2 = (87 – 61)2 = (26)2 = (20 + 6)2 = (20)2 + 62 + 2 × 20 × 6 = (400 + 36 + 240) = (436 + 240) = 676. Ex. 9. Find the square root of 4a2 + b2 + c2 + 4ab – 2bc – 4ac. (Campus Recruitment, 2010) Sol. 4 a 2 + b 2 + c 2 + 4 ab – 2bc – 4 ac = (2 a)2 + b 2 + (– c)2 + 2 × 2 a × b + 2 × b × (– c) + 2 × (2 a) × (– c) = (2 a + b – c )2 = (2 a + b – c ). Ex. 10. A is counting the numbers from 1 to 31 and B from 31 to 1. A is counting the odd numbers only. The speed of both is the same. What will be the number which will be pronounced by A and B together? (Campus Recruitment, 2010) Sol. The numbers pronounced by A and B in order are: A 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 B 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 Clearly both A and B pronounce the number 21 together. NUMBER SYSTEM 9 789 × 789 × 789 + 211 × 211 × 211 658 × 658 × 658 – 328 × 328 × 328 Ex. 11. Simplify: (i) (ii) 789 × 789 – 789 × 211 + 211 × 211 658 × 658 + 658 × 328 + 328 × 328 (789)3 + (211)3 a3 + b 3 Sol. (i) Given exp. = = , (where a = 789 and b = 211) (789)2 – (789 × 211) + (211)2 a2 – ab + b 2 = (a + b) = (789 + 211) = 1000. (658)3 – (328)3 a3 – b 3 (ii) Given exp. = = ,  (where a = 658 and b = 328) (658)2 + (658 × 328) + (328)2 a 2 + ab + b 2 = (a – b) = (658 – 328) = 330. (893 + 786)2 – (893 – 786)2 Ex. 12. Simplify:. (893 × 786) ( a + b )2 – ( a – b ) 2 4 ab Sol. Given exp. = (where a = 893, b = 786) = = 4. ab ab Ex. 13. Which of the following are prime numbers? (i) 241 (ii) 337 (iii) 391 (iv) 571 Sol. (i) Clearly, 16 > 241. Prime numbers less than 16 are 2, 3, 5, 7, 11, 13. 241 is not divisible by any of them. \ 241 is a prime number. (ii) Clearly, 19 > 337. Prime numbers less than 19 are 2, 3, 5, 7, 11, 13, 17. 337 is not divisible by any one of them. \ 337 is a prime number. (iii) Clearly, 20 > 391. Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. We find that 391 is divisible by 17. \ 391 is not a prime number. (iv) Clearly, 24 > 571. Prime numbers less than 24 are 2, 3, 5, 7, 11, 13, 17, 19, 23. 571 is not divisible by any one of them. \ 571 is a prime number. Ex. 14. If D stands for the operation 'adding first number to twice the second number', then find the value of (1 D 2) D 3. Sol. (1 D 2) D 3 = (1 + 2 × 2) D 3 = 5 D 3 = 5 + 2 × 3 = 5 + 6 = 11. Ex. 15. Given that 12 + 22 + 32 +..... + 102 = 385, then find the value of 22 + 42 + 62 +...... + 202. Sol. 22 + 42 + 62 +.... + 202 = 22 (12 + 22 + 32 +.... + 102) = 22 × 385 = 4 × 385 = 1540. Ex.16. Which of the following numbers is divisible by 3? (i) 541326 (ii) 5967013 Sol. (i) Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3. Hence, 541326 is divisible by 3. (ii) Sum of digits in 5967013 = (5 + 9 + 6 + 7 + 0 + 1 + 3) = 31, which is not divisible by 3. Hence, 5967013 is not divisible by 3. Ex. 17. What least value must be assigned to * so that the number 197*5462 is divisible by 9? Sol. Let the missing digit be x. Sum of digits = (1 + 9 + 7 + x + 5 + 4 + 6 + 2) = (34 + x). For (34 + x) to be divisible by 9, x must be replaced by 2. Hence, the digit in place of * must be 2. Ex. 18. Which of the following numbers is divisible by 4? (i) 67920594 (ii) 618703572 Sol. (i) The number formed by the last two digits in the given number is 94, which is not divisible by 4. Hence, 67920594 is not divisible by 4. (ii) The number formed by the last two digits in the given number is 72, which is divisible by 4. Hence, 618703572 is divisible by 4. 10 QUANTITATIVE APTITUDE Ex. 19. Which digits should come in place of * and $ if the number 62684*$ is divisible by both 8 and 5? Sol. Since the given number is divisible by 5, so 0 or 5 must come in place of $. But, a number ending with 5 is never divisible by 8. So, 0 will replace $. Now, the number formed by the last three digits is 4*0, which becomes divisible by 8, if * is replaced by 4. Hence, digits in place of * and $ are 4 and 0 respectively. Ex. 20. Show that 4832718 is divisible by 11. Sol. (Sum of digits at odd places) – (Sum of digits at even places) = (8 + 7 + 3 + 4) – (1 + 2 + 8) = 11, which is divisible by 11. Hence, 4832718 is divisible by 11. Ex. 21. Is 52563744 divisible by 24? Sol. 24 = 3 × 8, where 3 and 8 are co-primes. The sum of the digits in the given number is 36, which is divisible by 3. So, the given number is divisible by 3. The number formed by the last 3 digits of the given number is 744, which is divisible by 8. So, the given number is divisible by 8. Thus, the given number is divisible by both 3 and 8, where 3 and 8 are co-primes. So, it is divisible by 3 × 8, i.e. 24. Ex. 22. What are the values of M and N respectively if M39048458N is divisible by both 8 and 11, where M and N are single-digit integers? Sol. Since the given number is divisible by 8, it is obvious that the number formed  by the last three digits, i.e. 58N is divisible by 8, which is possible only when N =  4. Now, (sum of digits at even places) – (sum of digits at odd places) = (8 + 4 + 4 + 9 + M) – (4 + 5 + 8 + 0 + 3) = (25 + M) – 20 = M + 5, which must be divisible by 11. So, M = 6. Hence, M = 6, N = 4. Ex. 23. Find the number of digits in the smallest number which is made up of digits 1 and 0 only and is divisible by 225. Sol. 225 = 9 × 25, where 9 and 25 are co-primes. Clearly, a number is divisible by 225 if it is divisible by both 9 and 25. Now, a number is divisible by 9 if the sum of its digits is divisible by 9 and a number is divisible by 25 if the number formed by the last two digits is divisible by 25. \ The smallest number which is made up of digits 1 and 0 and divisible by 225 = 11111111100. Hence, number of digits = 11. Ex. 24. If the number 3422213pq is divisible by 99, find the missing digits p and q. Sol. 99 = 9 × 11, where 9 and 11 are co-primes. Clearly, a number is divisible by 99 if it is divisible by both 9 and 11. Since the number is divisible by 9, we have: (3 + 4 + 2 + 2 + 2 + 1 + 3 + p + q) = a multiple of 9 ⇒ 17 + (p + q) = 18 or 27 ⇒p+q=1...(i) or p + q = 10...(ii) Since the number is divisible by 11, we have: (q + 3 + 2 + 2 + 3) – (p + 1 + 2 + 4) = 0 or a multiple of 11 ⇒ (10 + q) – (7 + p) = 0 or 11 ⇒ 3 + (q – p) = 0 or 11 ⇒ q – p = – 3 or q – p = 8 ⇒p–q=3...(iii) or q – p = 8...(iv) Clearly, if (i) holds, then neither (iii) nor (iv) holds. So, (i) does not hold. Also, solving (ii) and (iii) together, we get: p = 6.5, which is not possible. Solving (ii) and (iv) together, we get: p = 1, q = 9. Ex. 25. x is a positive integer such that x2 + 12 is exactly divisible by x. Find all the possible values of x. x 2 + 12 x 2 12 12 Sol. = + =x+. x x x x Clearly, 12 must be completely divisible by x. So, the possible values of x are 1, 2, 3, 4, 6 and 12. NUMBER SYSTEM 11 Ex. 26. Find the smallest number to be added to 1000 so that 45 divides the sum exactly. Sol. On dividing 1000 by 45, we get 10 as remainder. \ Number to be added = (45 – 10) = 35. Ex. 27. What least number must be subtracted from 2000 to get a number exactly divisible by 17? Sol. On dividing 2000 by 17, we get 11 as remainder. \ Required number to be subtracted = 11. Ex. 28. Find the number which is nearest to 3105 and is exactly divisible by 21. Sol. On dividing 3105 by 21, we get 18 as remainder. \ Number to be added to 3105 = (21 – 18) = 3. Hence, required number = 3105 + 3 = 3108. Ex. 29. Find the smallest number of five digits which is exactly divisible by 476. Sol. Smallest number of 5 digits = 10000. On dividing 10000 by 476, we get 4 as remainder. \ Number to be added = (476 – 4) = 472. Hence, required number = 10472. Ex. 30. Find the greatest number of five digits which is exactly divisible by 47. 42735 Sol. Greatest number of 5 digits is 99999. 13 555555 On dividing 99999 by 47, we get 30 as remainder. 52 3 5 \ Required number = (99999 – 30) = 99969. 2 6 Ex. 31. When a certain number is multiplied by 13, the product consists entirely of fives. Find the 9 5 9 1 smallest such number. 4 5 Sol. Clearly, we keep on dividing 55555....... by 13 till we get 0 as remainder. 3 9 6 5 \ Required number = 42735. 6 5 Ex. 32. When a certain number is multiplied by 18, the product consists entirely of 2's. What is the × minimum number of 2's in the product? Sol. We keep on dividing 22222......... by 18 till we get 0 as remainder. 12345679 Clearly, number of 2's in the product = 9. 18 222222222 18 Ex. 33. Find the smallest number which when multiplied by 9 gives the product 4 2 as 1 followed by a certain number of 7s only. 3 6 Sol. The least number having 1 followed by 7s, which is divisible by 9, is 5 4 62 177777, as 1 + 7 + 7 + 7 + 7 + 7 = 36 (which is divisible by 9). 8 2 \ Required number = 177777 ÷ 9 = 19753. 7 2 1 0 2 Ex. 34. What is the unit's digit in the product? 9 0 81 × 82 × 83 ×........... 89? 122 Sol. Required unit's digit = Unit's digit in the product 1 × 2 × 3 ×.......× 9 = 0 1 0 8 [ 2 × 5 = 10] 142 153 72 126 Ex. 35. Find the unit's digit in the product (2467) × (341). 162 Sol. Clearly, unit's digit in the given product = unit's digit in 7153 × 172. 162 × Now, 74 gives unit digit 1. \ 7152 gives unit digit 1. \ 7153 gives unit digit (1 × 7) = 7. Also, 172 gives unit digit 1. Hence, unit digit in the product = (7 × 1) = 7. Ex. 36. Find the unit's digit in (264)102 + (264)103. Sol. Required unit's digit = unit's digit in (4)102 + (4)103. Now, 42 gives unit digit 6. \ (4)102 gives unit digit 6. (4)103 gives unit digit of the product (6 × 4) i.e., 4. Hence, unit’s digit in (264)102 + (264)103 = unit's digit in (6 + 4) = 0. Ex. 37. Find the total number of prime factors in the expression (4)11 × (7)5 × (11)2. Sol. (4)11 × (7)5 × (11)2 = (2 × 2)11 × (7)5 × (11)2 = 211 × 211 × 75 × 112 = 222 × 75 × 112. \ Total number of prime factors = (22 + 5 + 2) = 29. 12 QUANTITATIVE APTITUDE Ex. 38. What is the number of zeros at the end of the product of the numbers from 1 to 100? Sol. Let N = 1 × 2 × 3 ×......... × 100. Clearly, only the multiples of 2 and 5 yield zeros on multiplication. In the given product, the highest power of 5 is much less than that compared to 2. So, we shall find the highest power of 5 in N.  100   100  Highest power of 5 in N =  + = 20 + 4 = 24.  5   5 2  Hence, required number of zeros = 24. Ex. 39. What is the number of zeros at the end of the product 55 × 1010 × 1515 ×........ × 125125? Sol. Clearly, the highest power of 2 is less than that of 5 in N. So, the highest power of 2 in N shall give us the number of zeros at the end of  N. Highest power of 2 = Number of multiples of 2 + Number of multiples of 4 (i.e. 22) + Number of multiples of 8 (i.e. 2­3) + Number of multiples of 16 (i.e. 24) = [(10 + 20 + 30 +....... + 120) + (20 + 40 + 60 +........ + 120) + (40 + 80 + 120) + 80] = (780 + 420 + 240 + 80) = 1520. Hence, required number of zeros = 1520. Ex. 40. On dividing 15968 by a certain number, the quotient is 89 and the remainder is 37. Find the divisor. Dividend – Remainder 15968 – 37 Sol. Divisor = = = 179. Quotient 89 Ex. 41. A number when divided by 114, leaves remainder 21. If the same number is divided by 19, find the remainder. (S.S.C., 2010) Sol. On dividing the given number by 114, let k be the quotient and 21 the remainder. Then, number = 114 k + 21 = 19 × 6k + 19 + 2 = 19 (6k + 1) + 2. \ The given number when divided by 19 gives remainder = 2. Ex. 42. A number being successively divided by 3, 5 and 8 leaves remainders 1, 4 and 7 respectively. Find the respective remainders if the order of divisors be reversed. Sol. 3 x 5 y – 1 \ z = (8 × 1 + 7) = 15 ; y = (5 z + 4) = (5 × 15 + 4) = 79; 8 z – 4 x = (3y + 1) = (3 × 79 + 1) = 238. 1 – 7 Now, 8 238 5 29 – 6 3 5–4 1 – 2 \ Respective remainders are 6, 4, 2. Ex. 43. Three boys A, B, C were asked to divide a certain number by 1001 by the method of factors. They took the factors in the orders 13, 11, 7; 7, 11, 13 and 11, 7, 13 respectively. If the first boy obtained 3, 2, 1 as successive remainders, then find the successive remainders obtained by the other two boys B and C. Sol. 13 x 11 y – 3 \ z = 7 × 1 + 1 = 8, 7 z–2 y = 11 z + 2 = 11 × 8 + 2 = 90; 1 – 1 x = 13 y + 3 = 13 × 90 + 3 = 1173. Now, 7 1173 11   167 – 4 So, B obtained 4, 2 and 2 as successive remainders. 13   15 – 2 1–2 And, 11 1173 7 106 – 7 C obtained 7, 1 and 2 as successive remainders. 13   15 – 1 1–2 NUMBER SYSTEM 13 Ex. 44. In a division sum, the divisor is ten times the quotient and five times the remainder. If the remainder is 46, determine the dividend. 230 Sol. Remainder = 46 ; Divisor = 5 × 46 = 230 ; Quotient = = 23. 10 \ Dividend = Divisor × Quotient + Remainder = 230 × 23 + 46 = 5336. Ex. 45. If three times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as remainder. Also, if seven times the smaller number is divided by the larger one, we get 5 as quotient and 1 as remainder. Find the numbers. Sol. Let the larger number be x and the smaller number be y. Then, 3x = 4y + 3 ⇒ 3x – 4y = 3...(i) And, 7y = 5x + 1 ⇒ – 5x + 7y = 1...(ii) Multiplying (i) by 5 and (ii) by 3, we get: 15x – 20y = 15...(iii) and – 15x + 21y = 3...(iv) Adding (iii) and (iv), we get: y = 18. Putting y = 18 in (i), we get: x = 25. Hence, the numbers are 25 and 18. Ex. 46. A number when divided by 6 leaves remainder 3. When the square of the same number is divided by 6, find the remainder. Sol. On dividing the given number by 6, let k be the quotient and 3 the remainder. Then, number = 6k + 3. Square of the number = (6k + 3)2 = 36k2 + 9 + 36k = 36k2 + 36k + 6 + 3 = 6 (6k2 + 6k + 1) + 3, which gives a remainder 3 when divided by 6. Ex. 47. Find the remainder when 96 + 7 is divided by 8. Sol. (xn – an) is divisible by (x – a) for all values of n. So, (96 – 1) is divisible by (9 – 1), i.e. 8 ⇒ (96 – 1) + 8 is divisible by 8 ⇒ (96 + 7) is divisible by 8. Hence, required remainder = 0. Ex. 48. Find the remainder when (397)3589 + 5 is divided by 398. Sol. (xn + an) is divisible by (x + a) for all odd values of n. So, [(397)3589 + 1] is divisible by (397 + 1), i.e. 398 ⇒ [{(397)3589 + 1} + 4] gives remainder 4 when divided by 398 ⇒ [(397)3589 + 5] gives remainder 4 when divided by 398. Ex. 49. If 7126 is divided by 48, find the remainder. Sol. 7126 = (72)63 = (49)63. Now, since (xn – an) is divisible by (x – a) for all values of n, so [(49)63 – 1] or (7126 – 1) is divisible by (49 – 1) i.e. 48. \ Remainder obtained when (7)126 is divided by 48 = 1. Ex. 50. Find the remainder when (257166 – 243166) is divided by 500. Sol. (xn – an) is divisible by (x + a) for all even values of n. \ (257166 – 243166) is divisible by (257 + 243), i.e. 500. Hence, required remainder = 0. Ex. 51. Find a common factor of (127127 + 97127) and (12797 + 9797). Sol. (xn + an) is divisible by (x + a) for all odd values of n. \ (127127 + 97127) as well as (12797 + 9797) is divisible by (127 + 97), i.e. 224. Hence, required common factor = 224. Ex. 52. A 99–digit number is formed by writing the first 59 natural numbers one after the other as: 1234567891011121314...........5859 Find the remainder obtained when the above number is divided by 16. Sol. The required remainder is the same as that obtained on dividing the number formed by the last four digits i.e. 5859 by 16, which is 3. 14 QUANTITATIVE APTITUDE EXERCISE (OBJECTIVE TYPE QUESTIONS) Directions: Mark () against the correct answer in each (a) None (b) Only 1 of the following: (c) 1 and 2 (d) 2 and 3 1. What is the place value of 5 in 3254710? (CLAT, 2010) 12. Every rational number is also (a) 5 (b) 10000 (a) an integer (b) a real number (c) 50000 (d) 54710 (c) a natural number (d) a whole number 2. The face value of 8 in the number 13. The number p is (R.R.B., 2005) 458926 is (R.R.B., 2006) (a) a fraction (b) a recurring decimal (a) 8 (b) 1000 (c) a rational number (d) an irrational number (c) 8000 (d) 8926 14. 2 is a/an 3. The sum of the place values of 3 in the number (a) rational number (b) natural number 503535 is (M.B.A., 2005) (c) irrational number (d) integer (a) 6 (b) 60 (c) 3030 (d) 3300 15. The number 3 is 4. The difference between the place values of 7 and 3 (a) a finite decimal in the number 527435 is (b) an infinite recurring decimal (a) 4 (b) 5 (c) equal to 1.732 (c) 45 (d) 6970 (d) an infinite non-recurring decimal 5. The difference between the local value and the face 16. There are just two ways in which 5 may be expressed value of 7 in the numeral 32675149 is as the sum of two different positive (non-zero) (a) 5149 (b) 64851 integers, namely 5 = 4 + 1 = 3 + 2. In how many ways, 9 can be expressed as the sum of two different (c) 69993 (d) 75142 positive (non-zero) integers? (e) None of these (a) 3 (b) 4 6. The sum of the greatest and smallest number of five (c) 5 (d) 6 digits is (M.C.A., 2005) 17. P and Q are two positive integers such that (a) 11,110 (b) 10,999 PQ = 64. Which of the following cannot be the (c) 109,999 (d) 111,110 value of P + Q? 7. If the largest three-digit number is subtracted from (a) 16 (b) 20 the smallest five-digit number, then the remainder is (c) 35 (d) 65 (a) 1 (b) 9000 18. If x + y + z = 9 and both y and z are positive integers (c) 9001 (d) 90001 greater than zero, then the maximum value x can 8. The smallest number of 5 digits beginning with 3 take is (Campus Recruitment, 2006) and ending with 5 will be (R.R.B., 2006) (a) 3 (b) 7 (a) 31005 (b) 30015 (c) 8 (d) Data insufficient (c) 30005 (d) 30025 19. What is the sum of the squares of the digits from 9. What is the minimum number of four digits formed 1 to 9? by using the digits 2, 4, 0, 7 ? (P.C.S., 2007) (a) 105 (b) 260 (a) 2047 (b) 2247 (c) 285 (d) 385 (c) 2407 (d) 2470 20. If n is an integer between 20 and 80, then any of 10. All natural numbers and 0 are called the......... the following could be n + 7 except numbers. (R.R.B., 2006) (a) 47 (b) 58 (a) rational (b) integer (c) 84 (d) 88 (c) whole (d) prime 21. Which one of the following is the correct sequence 11. Consider the following statements about natural in respect of the Roman numerals: C, D, L and M? numbers: (Civil Services, 2008) (1) There exists a smallest natural number. (a) C > D > L > M (b) M > L > D > C (2) There exists a largest natural number. (c) M > D > C > L (d) L > C > D > M (3) Between two natural numbers, there is always 22. If the numbers from 1 to 24, which are divisible by a natural number. 2 are arranged in descending order, which number Which of the above statements is/are correct? will be at the 8th place from the bottom? (CLAT, 2010) NUMBER SYSTEM 15 (a) 10 (b) 12 34. If m, n, o, p and q are integers, then m (n + o) (p – q) (c) 16 (d) 18 must be even when which of the following is even? 23. 2 – 2 + 2 – 2 +.......... 101 terms =? (P.C.S., 2008) (a) m (b) p (a) – 2 (b) 0 (c) m + n (d) n + p (c) 2 (d) None of these 35. If n is a negative number, then which of the following 24. 98th term of the infinite series 1, 2, 3, 4, 1, 2, 3, 4, is the least? 1, 2,........ is (M.C.A., 2005) (a) 0 (b) – n (a) 1 (b) 2 (c) 2n (d) n2 (c) 3 (d) 4 36. If x – y = 8, then which of the following must be 25. If x, y, z be the digits of a number beginning from true? the left, the number is I. Both x and y are positive. (a) xyz (b) x + 10y + 100z II. If x is positive, y must be positive. (c) 10x + y + 100z (d) 100x + 10y + z III. If x is negative, y must be negative. 26. If x, y, z and w be the digits of a number beginning (a) I only (b) II only from the left, the number is (c) I and II (d) III only (a) xyzw 37. If x and y are negative, then which of the following (b) wzyx statements is/are always true? (c) x + 10y + 100z + 1000w I. x + y is positive. (d) 103x + 102y + 10z + w II. xy is positive. 27. If n and p are both odd numbers, which of the III. x – y is positive. following is an even number? (a) I only (b) II only (a) n + p (b) n + p + 1 (c) III only (d) I and III only (c) np + 2 (d) np 38. If n = 1 + x, where x is the product of four consecu- 28. For the integer n, if n3 is odd, then which of the tive positive integers, then which of the following following statements are true? is/are true? I. n is odd. II. n2 is odd. I. n is odd. II. n is prime. 2 III. n is even. III. n is a perfect square. (a) I only (b) II only (a) I only (b) I and II only (c) I and II only (d) I and III only (c) I and III only (d) None of these 29. If (n – 1) is an odd number, what are the two other 2 39. If x = y + 3, how does y change when x increases odd numbers nearest to it? 5 (a) n, n – 1 (b) n, n – 2 from 1 to 2? 5 (c) n – 3, n + 1 (d) n – 3, n + 5 (a) y increases from – 5 to – 2 30. Which of the following is always odd? 2 (b) y increases from to 5 (a) Sum of two odd numbers 5 (b) Difference of two odd numbers 5 (c) y increases from to 5 (c) Product of two odd numbers 2 5 (d) None of these (d) y decreases from – 5 to – 31. If x is an odd integer, then which of the following 2 is true? 40. If x is a rational number and y is an irrational number, then (a) 5x – 2 is even (b) 5x2 + 2 is odd (a) both x + y and xy are necessarily rational (c) 5x2 + 3 is odd (d) None of these (b) both x + y and xy are necessarily irrational 32. If a and b are two numbers such that ab = 0, then (c) xy is necessarily irrational, but x + y can be either (R.R.B., 2006) rational or irrational (a) a = 0 and b = 0 (b) a = 0 or b = 0 or both (d) x + y is necessarily irrational, but xy can be either (c) a = 0 and b ≠ 0 (d) b = 0 and a ≠ 0 rational or irrational 33. If A, B, C, D are numbers in increasing order and 41. The difference between the square of any two con- D, B, E are numbers in decreasing order, then which secutive integers is equal to one of the following sequences need neither be in (a) sum of two numbers a decreasing nor in an increasing order? (b) difference of two numbers (a) E, C, D (b) E, B, C (c) an even number (c) D, B, A (d) A, E, C (d) product of two numbers 16 QUANTITATIVE APTITUDE 42. Between two distinct rational numbers a and b, there 51. In the relation x > y + z, x + y > p and z < p, which exists another rational number which is (P.C.S., 2006) of the following is necessarily true? a b (Campus Recruitment, 2008) (a) (b) 2 2 (a) y > p (b) x + y > z a+b (c) y + p > x (d) Insufficient data ab (c) (d) 2 ( a − b) 4 2 52. If a and b are positive integers and = , 3.5 7 43. If B > A, then which expression will have the highest then (Campus Recruitment, 2010) value (given that A and B are positive integers)? (a) b > a (b) b < a (Campus Recruitment, 2007) (c) b = a (d) b ≥ a (a) A – B (b) AB 13 w 2 (c) A + B (d) Can't say 53. If 13 = , then (2 w) = ? (1 – w) 44. If 0 < x < 1, which of the following is greatest? (Campus Recruitment, 2009) (Campus Recruitment, 2007) 1 1 (a) (b) (a) x (b) x2 4 2 1 1 (c) 1 (d) 2 (c) (d) x x2 Directions (Questions 54–57): For a 5–digit number, without 45. If p is a positive fraction less than 1, then repetition of digits, the following information is available. 1 1 (B.B.A., 2006) (a) is less than 1 (b) is a positive integer p p (i) The first digit is more than 5 times the last digit. (c) p2 is less than p (ii) The two-digit number formed by the last two digits is the product of two prime numbers. 2 (d) – p is a positive number (iii) The first three digits are all odd. p (iv) The number does not contain the digits 3 or 0 46. If x is a real number, then x2 + x + 1 is and the first digit is also the largest. 3 (a) less than 54. The second digit of the number is 4 (a) 5 (b) 7 (b) zero for at least one value of x (c) 9 (c) always negative (d) Cannot be determined 3 (d) greater than or equal to 55. The last digit of the number is 4 1 1 1 1 (a) 0 (b) 1 47. Let n be a natural number such that + + + 2 3 7 n (c) 2 (d) 3 56. The largest digit in the number is is also a natural number. Which of the following (a) 5 (b) 7 statements is not true? (A.A.O. Exam, 2009) (c) 8 (d) 9 (a) 2 divides n (b) 3 divides n 57. Which of the following is a factor of the given (c) 7 divides n (d) n > 84 number? 48. If n is an integer, how many values of n will give (a) 2 (b) 3  16n2 + 7 n + 6  (c) 4 (d) 9 an integral value of  ?  n  58. The least prime number is (a) 2 (b) 3 (a) 0 (b) 1 (c) 4 (d) None of these (c) 2 (d) 3 49. If p > q and r < 0, then which is true? 59. Consider the following statements: (a) pr < qr (b) p – r < q – r 1. If x and y are composite numbers, then x + y is always composite. (c) p + r < q + r (d) None of these 2. There does not exist a natural number which is 50. If X < Z and X < Y, which of the following is neither prime nor composite. necessarily true? Which of the above statements is/are correct? I. Y < Z II. X2 < YZ (a) 1 only (b) 2 only III. ZX < Y + Z (c) Both 1 and 2 (d) Neither 1 nor 2 (a) Only I (b) Only II 60. The number of prime numbers between 0 and 50 is (c) Only III (d) None of these (a) 14 (b) 15 (c) 16 (d) 17 NUMBER SYSTEM 17 61. The prime numbers dividing 143 and leaving a 74. The sum of three prime numbers is 100. If one remainder of 3 in each case are of them exceeds another by 36, then one of the (a) 2 and 11 (b) 11 and 13 numbers is (c) 3 and 7 (d) 5 and 7 (a) 7 (b) 29 62. The sum of the first four primes is (c) 41 (d) 67 (a) 10 (b) 11 75. Which one of the following is a prime number? (c) 16 (d) 17 (a) 161 (b) 221 63. The sum of all the prime numbers from 1 to 20 is (c) 373 (d) 437 (a) 75 (b) 76 76. The smallest prime number, that is the fifth term of (c) 77 (d) 78 an increasing arithmetic sequence in which all the 64. A prime number N, in the range 10 to 50, remains four preceding terms are also prime, is unchanged when its digits are reversed. The square (a) 17 (b) 29 of such a number is (c) 37 (d) 53 (a) 121 (b) 484 77. The number of prime numbers between 301 and (c) 1089 (d) 1936 320 are 65. The remainder obtained when any prime number (a) 3 (b) 4 greater than 6 is divided by 6 must be (c) 5 (d) 6 (Campus Recruitment, 2007) 78. Consider the following statements: (a) either 1 or 2 (b) either 1 or 3 1. If p > 2 is a prime, then it can be written as (c) either 1 or 5 (d) either 3 or 5 4n + 1 or 4n + 3 for a suitable natural number n. 66. Which of the following is not a prime number? 2. If p > 2 is a prime, then (p – 1) (p + 1) is always (CLAT, 2010) divisible by 4. (a) 21 (b) 23 Of these statements, (c) 29 (d) 43 (a) (1) is true but (2) is false 67. Which of the following is a prime number? (CLAT, 2010) (b) (1) is false but (2) is true (a) 19 (b) 20 (c) (1) and (2) are false (c) 21 (d) 22 (d) (1) and (2) are true 68. Which of the following is a prime number? 79. What is the first value of n for which n2 + n + 41 (Campus Recruitment, 2008) is not a prime? (a) 115 (b) 119 (a) 1 (b) 10 (c) 127 (d) None of these (c) 20 (d) 40 69. Which of the following is a prime number? 80. Let X k = (p p 1 2.......p k) + 1, where p1, p2,........., pk are (R.R.B., 2006) the first k primes. (a) 143 (b) 289 Consider the following: (c) 117 (d) 359 1. Xk is a prime number. 70. The smallest value of natural number n, for which 2. Xk is a composite number. 2n + 1 is not a prime number, is 3. Xk + 1 is always an even number. (a) 3 (b) 4 Which of the above is/are correct? (c) 5 (d) None of these (a) 1 only (b) 2 only 71. The smallest three-digit prime number is (c) 3 only (d) 1 and 3 (a) 101 (b) 103 81. 6 × 3 (3 – 1) is equal to (CLAT, 2010) (c) 107 (d) None of these (a) 19 (b) 20 72. How many of the integers between 110 and 120 are prime numbers? (M.B.A., 2006) (c) 36 (d) 53 (a) 0 (b) 1 82. 1234 + 2345 – 3456 + 4567 =? (Bank Recruitment, 2010) (c) 2 (d) 3 (a) 4590 (b) 4670 (e) 4 (c) 4680 (d) 4690 73. Four prime numbers are arranged in ascending (e) None of these order. The product of first three is 385 and that of 83. 5566 – 7788 + 9988 =? + 4444 (Bank Recruitment, 2010) last three is 1001. The largest prime number is (a) 3223 (b) 3232 (R.R.B., 2006) (c) 3322 (d) 3333 (a) 9 (b) 11 (c) 13 (d) 17 (e) None of these 18 QUANTITATIVE APTITUDE 84. 38649 – 1624 – 4483 =? (Bank Recruitment, 2009) 97. From the sum of 17 and – 12, subtract 48. (E.S.I.C., 2006) (a) 32425 (b) 32452 (a) – 43 (b) – 48 (c) 34522 (d) 35422 (c) – 17 (d) – 20 (e) None of these 98. 60840 ÷ 234 =? 85. 884697 – 773697 – 102479 =? (Bank Recruitment, 2009) (a) 225 (b) 255 (a) 8251 (b) 8512 (c) 260 (d) 310 (c) 8521 (d) 8531 (e) None of these (e) None of these 86. 10531 + 4813 – 728 =? × 87 (Bank Recruitment, 2008) 99. 3578 + 5729 –? × 581 = 5821 (a) 168 (b) 172 (a) 3 (b) 4 (c) 186 (d) 212 (c) 6 (d) None of these (e) None of these 100. – 95 ÷ 19 =? 87. What is 394 times 113? (a) – 5 (b) – 4 (a) 44402 (b) 44522 (c) 0 (d) 5 (c) 44632 (d) 44802 101. 12345679 × 72 is equal to (e) None of these (a) 88888888 (b) 888888888 88. 1260 ÷ 14 ÷ 9 =? (Bank P.O., 2009) (c) 898989898 (d) 999999998 (a) 9 (b) 10 102. 8899 – 6644 – 3322 =? – 1122 (c) 81 (d) 810 (a) 55 (b) 65 (e) None of these (c) 75 (d) 85 89. 136 × 12 × 8 =? (Bank P.O., 2009) (e) None of these (a) 12066 (b) 13046 103. 74844 ÷? = 54 × 63 (Bank P.O., 2009) (c) 13064 (d) 13066 (a) 22 (b) 34 (e) None of these (c) 42 (d) 54 90. 8888 + 848 + 88 –? = 7337 + 737 (Bank P.O., 2009) (e) None of these (a) 1450 (b) 1550 104. 1256 × 3892 =? (c) 1650 (d) 1750 (a) 4883852 (b) 4888532 (e) None of these (c) 4888352 (d) 4883582 91. 414 ×? × 7 = 127512 (Bank P.O., 2009) (e) None of these (a) 36 (b) 40 105. What is 786 times 964? (Bank P.O., 2008) (c) 44 (d) 48 (a) 757704 (b) 754164 (e) None of these (c) 759276 (d) 749844 92. Product of 82540027 and 43253 is (e) None of these (a) 3570103787831 (b) 3570103787832 106. What is 348 times 265? (S.B.I.P.O., 2008) (c) 3570103787833 (d) 3570103787834 (a) 88740 (b) 89750 93. (46351 – 36418 – 4505) ÷? = 1357 (Bank P.O., 2009) (c) 92220 (d) 95700 (a) 2 (b) 3 (e) None of these (c) 4 (d) 6 107. (71 × 29 + 27 × 15 + 8 × 4) equals (S.S.C., 2007) (e) None of these (a) 2496 (b) 3450 94. 6 × 66 × 666 =? (Bank Recruitment, 2007) (c) 3458 (d) None of these (a) 263376 (b) 263763 108. ? × (|a| × |b|) = – ab (c) 263736 (d) 267336 (a) 0 (b) –1 (e) None of these (c) 1 (d) None of these 95. If you subtract – 1 from + 1, what will be the 109. (46)2 – (?)2 = 4398 – 3066 result? (R.R.B., 2006) (a) 16 (b) 28 (a) – 2 (b) 0 (c) 36 (d) 42 (c) 1 (d) 2 (e) None of these 96. 8 + 88 + 888 + 8888 + 88888 + 888888 =? 110. (800 ÷ 64) × (1296 ÷ 36) =? (a) 897648 (b) 896748 (a) 420 (b) 460 (c) 986748 (d) 987648 (c) 500 (d) 540 (e) None of these (e) None of these NUMBER SYSTEM 19 111. 5358 × 51 =? (c) 2704 (d) 2904 (a) 273258 (b) 273268 (e) None of these (c) 273348 (d) 273358 124. The value of 112 × 54 is 112. 587 × 999 =? (a) 6700 (b) 70000 (a) 586413 (b) 587523 (c) 76500 (d) 77200 (c) 614823 (d) 615173 125. Multiply 5746320819 by 125. 113. 3897 × 999 =? (a) 718,290,102,375 (b) 728,490,301,375 (a) 3883203 (b) 3893103 (c) 748,290,103,375 (d) 798,290,102,975 (c) 3639403 (d) 3791203 126. 935421 × 625 =? (e) None of these (a) 575648125 (b) 584638125 114. 72519 × 9999 =? (c) 584649125 (d) 585628125 (a) 725117481 (b) 674217481 127. (999)2 – (998)2 =? (R.R.B., 2008) (c) 685126481 (d) 696217481 (a) 1992 (b) 1995 (e) None of these (c) 1997 (d) 1998 115. 2056 × 987 =? 128. (80)2 – (65)2 + 81 =? (a) 1936372 (b) 2029272 (a) 306 (b) 2094 (c) 1896172 (d) 1923472 (c) 2175 (d) 2256 (e) None of these (e) None of these 116. 1904 × 1904 =? 129. (24 + 25 + 26)2 – (10 + 20 + 25)2 =? (a) 3654316 (b) 3632646 (a) 352 (b) 400 (c) 3625216 (d) 3623436 (c) 752 (d) 2600 (e) None of these (e) None of these 117. 1397 × 1397 =? 130. (65)2 – (55)2 =? (a) 1951609 (b) 1981709 (a) 10 (b) 100 (c) 18362619 (d) 2031719 (c) 120 (d) 1200 (e) None of these 131. If a and b be positive integers such that a2 – b2 = 19, 118. 107 × 107 + 93 × 93 =? then the value of a is (S.S.C., 2010) (a) 19578 (b) 19418 (a) 9 (b) 10 (c) 20098 (d) 21908 (c) 19 (d) 20 (e) None of these 132. If a and b are positive integers, a > b and (a + b)2 – 119. 217 × 217 + 183 × 183 =? (R.R.B., 2007) (a – b)2 > 29, then the smallest value of a is (a) 79698 (b) 80578 (a) 3 (b) 4 (c) 80698 (d) 81268 (c) 6 (d) 7 (e) None of these 133. 397 × 397 + 104 × 104 + 2 × 397 × 104 =? 120. 106 × 106 – 94 × 94 =? (a) 250001 (b) 251001 (c) 260101 (d) 261001 (a) 2400 (b) 2000 2 2 134. If (64) – (36) = 20 × x, then x =? (c) 1904 (d) 1906 (a) 70 (b) 120 (e) None of these (c) 180 (d) 140 121. 8796 × 223 + 8796 × 77 =? (e) None of these (a) 2736900 (b) 2738800 (c) 2658560 (d) 2716740 (489 + 375)2 – (489 – 375)2 135. =? (e) None of these (489 × 375) (a) 144 (b) 864 122. 287 × 287 + 269 × 269 – 2 × 287 × 269 =? (c) 2 (d) 4 (a) 534 (b) 446 (e) None of these (c) 354 (d) 324 (963 + 476)2 + (963 – 476)2 (e) None of these 136. =? (963 × 963 + 476 × 476) 123. {(476 + 424)2 – 4 × 476 × 424} =? (a) 2 (b) 4 (a) 2906 (b) 3116 20 QUANTITATIVE APTITUDE (c) 497 (d) 1449 146. Find the missing number in the following addition (e) None of these problem: 8 3 5 768 × 768 × 768 + 232 × 232 × 232 4 * 8 137. =? 786 × 768 – 768 × 232 + 232 × 232 + 9 * 4 (a) 1000 (b) 536 2 2 * 7 (c) 500 (d) 268 (a) 0 (b) 4 (e) None of these (c) 6 (d) 9 854 × 854 × 854 – 276 × 276 × 276 138. =? 147. What number should replace M in this multiplication 854 × 854 + 854 × 276 + 276 × 276 problem? (a) 1130 (b) 578 3 M 4 (c) 565 (d) 1156 × 4 (e) None of these 1 2 1 6 753 × 753 + 247 × 247 – 753 × 247 (a) 0 (b) 2 139. =? 753 × 753 × 753 + 247 × 247 × 247 (c) 4 (d) 8 1 1 148. If p and q represent digits, what is the maximum (a) (b) possible value of q in the statement (S.S.C., 2010) 1000 506 253 5p9 + 327 + 2q8 = 1114? (c) (d) None of these (a) 6 (b) 7 500 (c) 8 (d) 9 256 × 256 – 144 × 144 149. What would be the maximum value of Q in the 140. is equal to (S.S.C., 2010) 112 following equation? (a) 420 (b) 400 5P7 + 8Q9 + R32 = 1928 (c) 360 (d) 320 (a) 6 (b) 8 141. I f a = 1 1 a n d b = 9 , t h e n t h e v a l u e o f (c) 9 (d) Data inadequate  a + b + ab  2 2 (e) None of these  3 3  is (S.S.C., 2010) 150. What should come in place of * mark in the following  a –b  equation? 1 1 1*5$4 ÷ 148 = 78 (a) (b) 2 20 (a) 1 (b) 4 (c) 2 (d) 20 (c) 6 (d) 8 142. If a + b + c = 0, (a + b) (b + c) (c + a) equals (e) None of these (M.C.A., 2005) 151. If 6*43 – 46@9 = 1904, which of the following should (a) ab (a + b) (b) (a + b + c)2 come in place of *? (c) – abc 2 2 (d) a + b + c 2 (a) 4 (b) 6 (c) 9 (d) Cannot be determined 143. If a = 7, b = 5, c = 3, then the value of a2 + b2 + c2 – ab – bc – ca is (e) None of these 152. What should be the maximum value of Q in the (a) – 12 (b) 0 following equation? (c) 8 (d) 12 5P9 – 7Q2 + 9R6 = 823 144. Both addition and multiplication of numbers are (a) 5 (b) 6 operations which are (c) 7 (d) 9 (a) neither commutative nor associative (e) None of these (b) associative but not commutative 153. In the following sum, '?' stands for which digit? (c) commutative but not associative ? + 1? + 2? +? 3 +? 1 = 21? (d) commutative and associative (a) 4 (b) 6 145. Which of the following digits will replace the H (c) 8 (d) 9 marks in the following equation? (e) None of these 9H + H8 + H6 = 230 Directions (Questions 154–155): These questions are based (a) 3 (b) 4 on the following information: (c) 5 (d) 9 CBA + CCA = ACD, where A, B, C and D stand for (e) None of these distinct digits and D = 0. NUMBER SYSTEM 21 154. B takes the value (a) 495 (b) 545 (a) 0 (b) 5 (c) 685 (d) 865 (c) 9 (d) 0 or 9 166. A positive number, which when added to 1000, gives 155. C takes the value a sum which is greater than when it is multiplied (a) 0 (b) 2 by 1000. This positive integer is (c) 2 or 3 (d) 5 (a) 1 (b) 3 156. A 3–digit number 4a3 is added to another 3–digit (c) 5