Quantitative Aptitude Preparation Numbers. Prepared by: MS. RUPAL PATEL Assistant Professor CMPICA, CHARUSAT

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1 Quantitative Aptitude Preparation Numbers Prepared by: MS. RUPAL PATEL Assistant Professor CMPICA, CHARUSAT

2 Numbers Numbers In Hindu Arabic system, we have total 10 digits. Namely, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Number is a group of digits called numeral. Places of each digit in a numeral Ten Crores Crores Ten Lakhs Lakhs Ten Thousands Thousand Hundreds Tens Units

3 Face value and Place Value

4 Types of Numbers Numbers Quick Description Symbol Natural {1, 2, 3,.} N Whole {0,1, 2, 3, } W Integers {..-3, -2, -1, 0, 1, 2, 3, } Z Rational Q Irrationals Not Rational Real All Rational and Irrational R Imaginary I Complex C

5 Even and Odd Numbers The integers which are divisible by 2 are called even numbers. E.g. 0, 2, 4, 6, 8,, etc. The integers which are not divisible by 2 are called odd numbers. E.g. 1, 3, 5, 7, 9,.., etc.

6 Prime Numbers A counting number which has only two factors: 1 and itself is called prime number. E.g. Prime numbers between 1 and 100 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 IMP: The only even prime number is 2.

7 Composite Numbers and Co-primes The natural numbers which are not prime are called composite numbers. E.g. 4, 6, 8, 9, 10, 12, 14, 15, 16, etc. Two natural numbers a and b are said to be co-prime if their HCF is 1. Means both a and b has no common factor. E.g. (2,3), (4,5), (7,9), (8,11), etc. IMP: 1 is neither prime nor a composite number

8 Important

9 Modulus or Absolute value of a Real Number

10 Test of Divisibility of Numbers Div. Rule Example (s) 2 If the last digit is an even number or zero(0) 3 If the sum of the digits divisible by 3 4 If the last two digits divisible by 4 or it ends with 00 84, 138, 2, = 15 is divisible by , since 60 is divisible by last 2 digits 00 5 If the last digit is 0 or 5 865, 1705, 25, 4270, If the number is divisible by both 2 and 3 8 If the last three digits divisible by 8 or it ends with last digit 0 so divisible by =21 is divisible by , since 976 is divisible by last 3 digits 000

11 Test of Divisibility of Numbers Div. Rule Example (s) 9 If the sum of the digits is divisible by = If the last digit is 0 730, 20, If the difference = sum of digits of (odd places - even places) is 0 or a multiple of 11 7 and 13 If the difference of the number formed by the last 3 digits and the number formed by the rest digits divisible by 7 or 13 respectively Odd sum = 6+7+5=18 Even Sum = = 7 Difference = 18-7 = , = 49 is divisible by = = 884 is divisible by 13

12 General Properties of Divisibility 1. If a number A is divisible by another number B, then any number divisible by A, will also be divisible by B and all the factors of B. E.g. 84 is divisible by is divisible by 84, it will also be divisible by 6 and also factors of 6, i.e. 2 and 3

13 General Properties of Divisibility 2. If two numbers A and B are divisible by a number p, then their sum A+B is also divisible by p E.g. 225 and 375 are divisible by 5. Then, = 600 is also divisible by If two numbers A and B are divisible by a number p, then their difference A-B is also divisible by p E.g. 126 and 507 are divisible by 3. Then, = 381 is also divisible by 3.

14 Must Memorize Tables of 1 to 20

15 Squares of 1 to 100 Must Memorize squares of 1 to 30

16 Test of Prime Numbers

17 E.g. Test of Prime Numbers I = 144 > 137 prime numbers < 12 are 2, 3, 5, 7, 11 None of them divides 137. Thus it is a prime. II = 225 > 203 Prime numbers < 15 are 2, 3, 5, 7, 11, is divisible by 7. Thus, it is not a prime.

18 Divisibility and Remainder

19 Methods to find Completely Divisible by another number Least number that can be subtracted to make it exactly divisible Answer = Remainder Least number that can be added to make it exactly divisible Answer = Divisor - Remainder

20 E.g. Find the least number, that must subtracted from and added to a given number 5029, to make it exactly divisible by 17. Also find the numbers in each case. Sol n 17) 5029 ( Least number subtracted = remainder = 14 i.e = 5015 Least number added = divisor remainder = = 3 i.e = 5032

21 Greatest n-digit and Least n-digit number exactly divisible by a number Greatest n-digit number exactly divisible by a divisor d Answer = Greatest n-digit no. Remainder Least n-digit number exactly divisible by a divisor d Answer = Least n-digit No. + (Divisor Remainder)

22 E.g. Find the Sol n (a) Greatest 3-digit number divisible by 13 (b) Least 3-digit number divisible by 13 13) 999 (76 13) 100 ( Greatest 3-digit number = = 988 Least 3-digit number = (13 9) = 104

23 Remainder Rules No. Type of Problem Rule 1 When a number is divided by x then leaves remainder R, when same number is divided by y, what will be the remainder? 2 If two different numbers a and b are divided by the same divisor leaves remainder r1 and r2, respectively, then their sum (a+b) if divided by same divisor will leave remainder R 3 When two numbers after being divided by the same divisor leave the same remainder, then the difference of those two numbers must be exactly divisible by the same divisor. R = (r 1 +r 2 ) divisor If R becomes negative, then R = r 1 +r 2 1. Find the difference of given two numbers 2. Find the factors of it. 3. Find the divisor according to n-digits asked

24 Remainder Rules No. Type of Problem Rule 4 If a given number is divided successively by the different factors of the divisor leaving remainders r 1, r 2 and r 3, respectively, then the true remainder is 5 When (x + 1) n is divided by x, then remainder is 6 When (x 1) n is divided by x, then remainder is True Remainder = (r 1 ) + (r 2 x d 1 ) +(r 3 x d 1 x r 2 ) Remainder is always 1. When x and n are naturals When n = even no, R = 1, When n = odd no, R = (x 1)

25 Sum Rules on Natural Numbers

27 Algebraic Formulas

28 Algebraic Formulas

29 Last digit (Unit s digit) of (xyz) n The given number is (xyz) n then last digit of a number is z. Based on this z, we, decide the last digit of a number when its power is calculated. Value of z Divide n Value of Remainder Last digit of z n 0, 1, 5, or , 1, 5, or 6, respectively 4 or 9 By , 3, 7 or 8 By z z 2 z z 2 z 3 z 4

31 Number of zeroes at the End of Product There are 2 reasons of 0 s at the end of product 1. If there is any 0 at the end of the number. E.g. 7 x 20 = If 5 or multiple of 5 multiplied by any even no. E.g. 45 x 12 = 540 Steps 1. Find the factors of given number 2. Count number of 2 s and 5 s 3. Number of 0 s = Min (No. of 2 s and No. of 5 s)

32 E.g. Find the number of zeroes at the end of the product of 15 x 32 x 25 x 22 x 40 x 75 x 98 x 112 x 125 = (5x3) x (2 5 ) x (5 2 ) x (11x2) x (2 3 x5) x (5 2 x3) x (2x7 2 ) x (2 4 x7) x (5 3 ) No. of 2 s = ( ) = 2 14 = 14 No. of 5 s = ( ) = 5 9 = 9 Min(9,14) = 9 Hence, There are total 9 zeroes at the end of the product.

33 Practice sums 1. The difference between the local value and face value of 6 in the numeral is (a) 6888 (b) 6886 (c) 5940 (d) None 2. The difference between place value s of two 4 in the numeral is (a) 0 (b) (c) (d) None

34 Practice sums 3. Find the least value of * for which 7*5462 is divisible by 9. (a) 2 (b) 0 (c) 3 (d) 5 4. Find the least value of * for which 4832*18 is divisible by 11. (a) 1 (b) 5 (c) 7 (d) 9 5. Show that is divisible by 24.

35 Practice sums 6. What least number must be subtracted from 1672 to obtain a number which is completely divisible by 17. (a) 3 (b) 4 (c) 6 (d) 5 7. What least number must be added to 2010 to obtain a number which is completely divisible by 17. (a) 2 (b) 4 (c) 7 (d) 9

36 Practice sums 8. On dividing by a certain number, we get 76 as quotient and 13 as remainder, What is the divisor? (a) 162 (b) 165 (c) 163 (d) On dividing a certain number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder? (a) 1 (b) 3 (c) 5 (d) 11

37 Practice sums 10. What is the unit digit in the product (684 x 759 x 413 x 676) (a) 4 (b) 8 (c) 1 (d) 3 11.What is the unit digit in the product (3547) 153 x (251) 72? (a) 1 (b) 2 (c) 4 (d) What is the unit digit in (264) (264) 103 (a) 4 (b) 0 (c) 3 (d) 1

38 Practice sums 13. Find the total number of prime factors in the product {(4) 11 x (7) 5 x (11) 2 }? (a) 22 (b) 25 (c) 29 (d) A number when 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. (a) 4,2,6 (b) 6,4,2 (c) 2,4,6 (d) 6,2,4

39 Practice sums 15. Find the remainder when 2 31 is divided by 5. (a) 2 (b) 3 (c) 9 (d) How many natural numbers between 17 and 80 are divisible by 6? (a) 12 (b) 16 (c) 11 (d) ( ) =? (a) 666 (b) 667 (c) 678 (d) Find the sum of all even natural numbers less than 75. (a) 1407 (b) 1466 (c) 1403 (d) 1406

40 Practice sums 19. What is the unit digit in (a) 1 (b) 5 (c) 7 (d) What is the unit digit in the product (3 65 x 6 59 x 7 71 )? (a) 1 (b) 2 (c) 4 (d) What is the unit digit in {(6374) 1793 x (625) 317 x (341) 491? (a) 0 (b) 2 (c) 3 (d) 5

41 Practice sums x 999 (a) (b) (c) (d) x 1904 (a) (b) (c) (d) x x 93 (a) (b) (c) (d) 21908

42 Practice sums 25.Which one of following numbers is exactly divisible by 11? (a) (b) (c) (d) The sum of first 45 natural numbers is (a) 1035 (b) 1280 (c) 2070 (d) The sum of even numbers between 1 and 31 is (a) 6 (b) 128 (c) 240 (d) 512

43 Practice sums 28.What smallest number should be added to 4456 so that it is completely divisible by 6? (a) 4 (b) 3 (c) 2 (d) The largest 4 digit number exactly divisible by 48 is (a) 9944 (b) 9768 (c) 9988 (d) The difference of two numbers is On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the smaller number? (a) 240 (b) 270 (c) 295 (d) 360

Multiples and Divisibility

Multiples and Divisibility A multiple of a number is a product of that number and an integer. Divisibility: A number b is said to be divisible by another number a if b is a multiple of a. 45 is divisible