Mathematics Chapter 3: Playing With Numbers - Pinkz Public

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Name: _____________________

Grade : VI

Subject : Mathematics

Chapter 3: Playing With Numbers

1. Which of the following is not co-prime? a. 8, 10 b. 11, 12 c. 1, 3 d. 31, 33

2. The number of common prime factors of 75, 60, 105 is a. 2 b. 3 c. 4 d. 5

3. The number of distinct prime factors of the largest 4-digit number is a. 2 b. 3 c. 5 d.11

4. Which of the following number is divisible by 8? a. 293 b. 1205 c. 1648 d. 2063

5. The number of distinct prime factors of the smallest 5-digit number is a. 2 b. 4 c. 6 d. 8

6. The HCF of 144 and 160 is a. 24 b. 15 c. 9 d. None of these

7. A number is divisible by 5 and 6. It may not be divisible by a. 10 b. 15 c. 30 d. 85

8. The LCM of 12, 24, 32 is a. 92 b. 86 c. 27 d. 96

9. The largest number, which always divides the sum of any pair of consecutive odd numbers is a. 2 b. 4 c. 6 d.8

10. Which of the following is a prime number? a. 143 b. 131 c. 147 d. 161

11. The number of primes between 16 to 80 and 90 to 100 is a. 20 b. 18 c. 17 d. 16

12. If the number 7254*98 is divisible by 22, the digit at * is a. 1 b. 2 c. 6 d. 0

I. Multiple Choice Questions

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1. a 2. a 3. b 4. c 5. a 6. d 7. d 8. d 9. d 10. b 11. c 12. c

1. Number of even numbers between 58 and 80 is a. 10 b. 11 c. 12 d. 13

2. Which of the following statements is not true? a. The HCF of two distinct prime numbers is 1 b. The HCF of two co-prime numbers is 1 c. The HCF of two consecutive even numbers is 2 d. The HCF of an even and an odd number is even

3. The number of distinct prime factors of the largest 4-digit number is a. 2 b. 3 c. 5 d. 11

4. The sum of the prime factors of 1729 is: a. 13 b. 19 c. 32 d. 39

5. The greatest number which always divides the product of the predecessor and successor of an odd natural number other then 1, is a. 6 b. 4 c. 16 d. 8

6. Which of the following numbers is divisible by 11? a. 1011011 b. 1111111 c. 2222222 d. 3333333

7. LCM of 10, 15 and 20 is a. 30 b. 60 c. 90 d. 180

8. LCM of two numbers is 180. Then which of the following is not the HCF of the numbers? a. 45 b. 60 c. 75 d. 90

9. Which of the following is the factor of every number? a. -1 b. 0 c. 1 d. any number

10. How many factors does 36 have? a. 5 b. 6 c. 7 d. 9

11. Which of the following is the smallest prime number? a. 1 b.2 c. 3 d. 4

12. How many prime numbers are there between 1 & 20? a. 7 b. 8 c. 9 d. 10

13. Which of the following is the smallest composite number? a. 2 b. 3 c. 4 d. 5

II. Multiple Choice Questions

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14. Which of the following is the HCF of the number 20, 28 & 36?

a. 20 b. 36 c. 4 d. 5 15. Which of the following is the LCM of the numbers 20, 30 & 60?

a. 20 b. 30 c. 60 d. 3600 16. Which of the following number is a perfect number:

a. 4 b. 6 c. 8 d. 12 17. The number of primes between 90 and 100 is:

a. 0 b. 1 c. 2 d. 3 18. Three numbers are in the ratio 1: 2: 3 and their HCF is 6, the numbers are:

a. 4, 8, 12 b. 5, 10, 15 c. 6, 12, 18 d. 10, 20, 30 19. The LCM of 24, 36 and 40 is:

a. 4 b. 90 c. 360 d. 720 1.a 2. d 3. d 4. b 5. a 6. c 7. b 8. c 9. c 10. b 11. b 12. b 13. c 14. c 15. b 16. b 17. b 18. c 19. c

1. Which of the following numbers is not a factor of 36? a. 2 b. 4 c. 18 d. 8







8. Which of the following number is not a factor of 18? a. 2 b. 3 c. 6 d. 8

III. Multiple Choice Questions

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9. Which of the following number is not a multiple of 6?

a. 12 b. 21 c. 24 d. 36 10. Which of the following number is not a multiple of 5?

a.15 b. 20 c. 30 d. 33 11. Which of the following number is not a multiple of 9?

a. 18 b. 36 c. 81 d. 75 12. The smallest prime number is

a. 1 b. 2 c. 3 d. 4 13. The smallest composite number is

a. 1 b. 2 c. 3 d. 4 14. The prime number which is even is

a. 2 b. 3 c. 5 d. 13 15. 1 is

a. a prime number b. a composite number c. neither prime nor composite d. an even number

16. The smallest odd number is a. 1 b. 2 c. 3 d. 4

17. The smallest even number is a. 1 b. 2 c. 3 d. 4

18. The least prime number between 1 and 10 is a. 2 b. 5 c. 3 d. 7

19. The greatest prime number between 1 and 10 is a. 7 b. 5 c. 3 d. 2

20. Which of the following numbers is Prime? a. 21 b. 12 c. 17 d. 39

21. Which of the following numbers is composite? a. 19 b. 23 c. 6 d. 29

22. Which of the following statements is true? a. The product of two even numbers is always even b. The sum of three odd numbers is even. c. All prime numbers are odd d. Prime numbers do not have any factors.

23. Which of the following statements is false? a. All even numbers are composite numbers. b. If an even number is divided by 2, the quotient may be odd or even. c. The sum of two odd numbers and one even number is even d. Sum of two prime numbers is not always even.

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24. 128 is divisible by

a. 2 b. 3 c. 5 d. 10 25. 275 is divisible by





a. 10 b. 6 c. 3 d. 11 30. The greatest common factor of 8 and 20 is

a. 2 b. 1 c. 4 d. 8 31. The greatest common factor 9 and 15 is

a. 3 b. 6 c. 9 d. 15 32. The common factor of 4 and 15 is

a. 1 b. 2 c. 3 d. 5 33. Which of the following pairs of number are not co-prime?

a. 7, 15 b. 12, 49 c. 18, 23 d. 12, 21 34. Which of the following Pairs of number are co-primes?

a. 30,415 b. 17, 68 c. 16, 81 d. 15, 100 35. The greatest common factor of 4,8 and 12 is

a. 2 b. 1 c. 4 d. 8 36. The greatest common factor of 5, 15 and 25 is



a. 1 b. 3 c. 5 d. 15 39. The smallest common factor of 4, 12 and 16 is

a. 1 b. 2 c. 4 d. 8 40. The HCF of 8 and 12 is


a. 3 b. 6 c. 12 d. 24. 42. The HCF of 27 and 63 is

a. 3 b. 6 c. 9 d. 18

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43. The HCF of 36 and 84 is


a. 2 b. 3 c. 5 d. 6 45. The LCM of 5 and 20 is




a. 3 b. 9 c. 5 d. 45. 49. The LCM of 5 and 6 is

a. 10 b. 20 c. 30 d. 60 1.d 2. d 3. c 4. c 5. d 6. d 7. d 8. d 9. b 10.d 11. d 12. b 13. d 14. a 15. c 16.a 17.b 18.a 19.a 20. c 21. c 22. a 23.a 24.a 25.a 26.a 27.a 28.a 29.a 30.c 31.a 32.a 33.d 34.c 35.c 36.b 37.a 38.d 39.a 40.d 41.c 42.c 43.d 44.d 45.d 46.c 47.d 48.d 49.c

1. Numbers having more than two factors are called ___________ numbers. 2. A number for which the sum of all its factors is equal to twice the number is

called a _______________ number. 3. 15 is a multiple of _____________ and ______________. 4. The common factor of 25 and 30 is __________________. 5. The LCM of 12 and 15 is ________________. 6. The HCF of 18 and 36 is ________________. 7. If the sum of the digits in a number is a _______________ of 3, then the

number is divisible by 3. 8. If the difference between the sum of digits at odd places (from the right) and

the sum of digits at even places (from the right) of a number is either 0 or divisible by _______________, then the number is divisible by 11.

9. The greatest prime number between 60 and 70 is _______________. 10. The smallest prime number between 80 and 90 is _______________. 11. The LCM of two or more given numbers is the lowest of their common _______.

I. Fill in the Blanks

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12. The HCF of two or more given numbers is the highest of their common _______. 13. The greatest 2-digit prime number is _____________.

1. Composite 2. Perfect 3. 3,5 4.5 5.60 6. 18 7.multiple 8.11 9. 67 10. 83 11.Multiples 12.Factor

of them 13.97

1. A number is a ___________ of each of its factors. 2. ________ is a factor of every number. 3. The number of factors of a prime number is ____________. 4. The numbers having more than two factors are called ___________ numbers. 5. Two numbers having only 1 as a common factor are called _______ numbers. 6. Numbers of primes between 1 to 100 is ___________. 7. If a number has ___________ in ones place, then it is divisible by 10. 8. A number is divisible by 5, if it has ________ or _______ in its ones place. 9. A number is divisible by _______ if it has any of the digits 0, 2, 4, 6 or 8 in its

ones place. 10. The HCF of two consecutive odd numbers is _____________. 11. A number having only two factors is called a ____________.

1. Multiple 2. 1 3. 2 4. Prime 5. Co-prime 6. 25 7. 0 8. 0, 5 9. 2 10. 1 11. Prime Number

1. The difference of two consecutive whole number a. odd 2. The product of two non zero consecutive whole numbers

b. 0

3. Quotient when zero is divided by another non-zero whole number

c. 3

4. 2 added the three, to the smallest whole number d. 1 5. Smallest odd prime number e. 6 f. even

1. d 2. f 3. b 4. e 5. c

II. Fill in the Blanks

I. Match the following

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1. Factor of 27 a. 64 2. Divisible by 11 b. 9 3. A multiple of 8 c. 1 4. HCF of 40 and 45 d. 5 5. Perfect number e. 81345 6. Neither a prime nor a composite number f.6

1. b 2. e 3. a 4. d 5. f 6. c

1. A number with three or more digits is divisible by 6, if the number formed by its last two digits (i.e. ones and tens) is divisible by 6.

2. A number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8.

3. If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 9.

4. 2 is the only even prime number. 5. Two consecutive even prime numbers are known as twin primes. 6. Two co-primes numbers are always prime numbers. 7. All numbers which are divisible by 4 may not be divisible by 8. 8. The highest common factor of two or more numbers is greater than their lowest

common multiple. 9. 27 is a perfect number. 10. It is possible to find the greatest prime number. 11. LCM of two or more numbers may be one of numbers. 12. HCF of two or more numbers may be one of the numbers. 13. The LCM of two co-prime numbers is equal to the product of the numbers.

1. False 2. True 3. False 4. True 5. False 6. False 7. True 8. False 9. False 10. False 11. True 12. True 13. True

II. Match the following

I. True or False

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1. Every multiple of a number is greater than or equal to the number.

2. The number of multiples of a given number is finite. 3. Every number is a multiple of itself. 4. Sum of two consecutive odd numbers is always divisible by 4. 5. If a number divides three numbers exactly, it must divide their sum exactly. 6. If a number exactly divides the sum of three numbers, it must exactly divide

the numbers separately. 7. If a number is divisible both by 2 and 3, then it is divisible by 12. 8. A number with three or more digits is divisible by 6, if the number formed by its

last two digits. (i.e., one and tens) is divisible by 6. 9. A number with 4 or more digits is divisible by 8, if the number formed by the

last three digits is divisible by 8. 10. If the sum of the digits of a number is divisible by 3, then the number itself is

divisible by 9. 11. All numbers which are divisible by 4 may not be divisible by 8. 12. The Highest common Factor of two or more numbers is greater than their

Lowest Common Multiple. 13. LCM of two more numbers is divisible by their HCF. 14. LCM of two numbers is 28 and their HCF is 8. 15. LCM of two or more numbers may be one of the numbers. 16. HCF of two or more numbers may be one of the numbers. 17. Every whole number is the successor of another whole number. 18. Sum of two whole numbers is always less than their product. 19. If the sum of two distinct whole numbers is odd, then their difference also must

be odd. 20. Any two consecutive numbers are co- prime. 21. If the HCF of two numbers is one of the numbers, then their LCM is the other

number. 22. The HCF of two numbers is smaller than the smaller of the numbers. 23. The LCM of two numbers is greater than the larger of the numbers. 24. The LCM of two co-prime numbers is equal to the product of the numbers. 25. A number and its successor are always co-prime. 26. Two consecutive numbers cannot be both primes. 27. The sum of the two consecutive odd numbers is always divisible by 4. 28. A number is divisible by 18, if it is divisible by both 3 and 6.

II. True or False

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1. True 2. False 3. True 4. True 5. True 6. False 7. False 8. False 9. True 10. False 11. True 12. False 13. True 14. False 15. True 16. True 17. False 18. False 19. True 20. True 21. True 22. False 23. False 24. True 25. True 26. False 27. True 28. False

1. Write all the factors of 24. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

2. Write the smallest multiple of a number.

The smallest multiple of a number is number itself 3. Write the greatest multiple of a number.

We cannot write the greatest multiple of a number. 4. Write all the multiple of 6.

We know that, the number of multiples of a number are infinite, So, we cannot write all the multiples of 6.

5. What is the perfect number? Give an example. A number is called perfect number, if it is equal to the sum of all its factors (except the number itself? e. g. Factors of 6 = 1, 2, 3 and 6 6 = 1 + 2 + 3 = 6

6. Write the greatest and the smallest prime number between 10 and 20. Greatest prime number = 19 Smallest prime number = 11

7. Write the smallest prime number. 2 is the smallest prime number.

8. Write some examples of composite numbers. Numbers having more than two factors are called composite numbers. e. g. 4, 6, 8, 9…... etc.

I. Very Short Answer Type Questions

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9. Which of the following are composite numbers?

7, 13, 16 Given numbers are 7, 13, 16 Factors of 7 = 1, 7 Factors of 13 = 1, 13 Factors of 16 = 1,2,4,8,16 ∴ 16 is a composite number.

10. Write the common factors of 6 and 10. Factors of 6 = 1, 2, 3, 6 Factors of 10= 1, 2, 5, 10 Common factors of 6 and 10 = 1, 2.

11. Write the first two common multiples of 5 and 15. Multiples of 5 = 5, 10, 15, 20, 25, 30… Multiples of 15 = 15, 30… First two common multiples = 15, 30.

12. A number is divisible by 4 and 3. By which other number will that number be always divisible? The number is divisible by 4 X 3 = 12.

13. A number is divisible by 20. By what other number will that number be divisible? Factors of 20 = 1, 2, 4, 5, 10, 20. ∴ The given number is divisible by 1, 2, 4, 5, 10.

14. Find the prime factorization of 40. Here, we have 40

∴ 40= 1x2x2x2x5

15. Write the highest common factor of 6 and 15. Factors of 6 = 1, 2, 3, 6

63)693(11 Factors of 15 = 1, 3, 5, 15 63 Common factors = 1, 3 63 Highest common factor = 3. 63 0

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16. Write the first three multiples of each of the following:

a. 11 b. 12 a. First three multiples of 11 = 11, 22, 33 b. First three multiples of 12 = 12, 24, 36

17. What is factor of a number? A number, which divides the given number completely, is called its factor.

18. Find the HCF of 16 and 20. For finding the HCF of 16 and 20, we use division method. ____ 16)20(1 16 ___ 4) 16(4 16 0 Hence, HCF of 16 and 20 is 4.

19. Write all the prime numbers between a. 10 and 15 b. 20 and 35 a. Prime numbers between 10 and 15 = 11, 13 b. Prime numbers between 20 and 35 = 23, 29, 31.

20. If 3 and 10 are two co-prime numbers. Find their LCM. We know that, the LCM of two co-prime numbers is equal to their product. ∴ The LCM of 3 and 10 = 3x 10 = 30.

1. Write the missing numbers for the following factor tree for 60.

Hence, the missing numbers are 2, 3, 5 and 2.

II. Very Short Answer Type Questions

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2. Write the missing numbers for the following factor tree for 60.

Hence, the missing numbers are 30, and 2.

3. Determine prime factors of 20570.

∴ 20570 = 2x5 x 11x 11 x 17

4. Is 35 a multiple of 6? No, as 35 = 5 X 7

5. Write the factors of 15 and show how they pair up.

Factors of 15 are 1, 3, 5, 15. So, 1 X 15 = 15, 3 X 5 = 15

6. Find the HCF of 156 and 208. HCF of 156 and 208. _____ 156) 208(1 -156 0 ∴ HCF = 52.

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7. Find LCM of 60 and 40.

LCM of 60 and 40

LCM = 23 X 3 X 5 = 120

8. Can two numbers have 18 as their HCF and 162 as their LCM? Give reasons. HCF= 18, LCM = 162. Since 162 is a multiple of 18, as 162 = 9 X 18 ∴ Yes two numbers can have 18 as HCF and 162 as LCM.

9. Using each of the digits 1, 2, 3 and 4 only once, determine the smallest 4-digits number divisible We know that, a no. is divisible by 4, if the last two digits of the number is divisible by 4. Hence, the required no. is 1324.

1. Find all the multiples of 9 upto 100. Multiples of 9 upto 100 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.

2. What is the sum of any two (i) odd numbers? (ii) Even numbers? (i) Sum of two odd numbers is always an even number. (ii) Sum of two even numbers is an even number only.

3. A number is divisible by both 5 and 12. By which other number will that number be always divisible? The number would be = 5 X 12 = 60.

4. A number is divisible by 12. By what other numbers will that number be divisible? If a number is divisible by 12, then other numbers by which it is divisible would be factors of 12. Factors of 12 are 1, 2, 3, 4, 6 So, the other numbers are 1, 2, 3, 4, 6

III. Very Short Answer Type Questions

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5. Write the smallest 5-digit number and express it in the form of its prime

factors. Smallest five digit number is 10, 000 Prime factors of 10, 000 = 2 X 2 X 2 X 2 X 5 X 5 X 5 X 5.

6. I am the smallest number, having four different prime factors. Can you find me? Smallest four prime numbers are 2, 3, 5, 7. Thus the smallest number is 2 X 3 X 5 X 7. Smallest Number = 2 X 3 X 5 X 7. = 6 X 5 X 7 = 30 X 7 = 210.

7. Which factors are not included in the prime factorization of a composite number? In prime factorization of a composite number, factor 1 and the number itself are not included.

8. Make a list of seven consecutive numbers, none of which is Prime. Seven consecutive numbers are 90, 91, 92, 93, 94, 95, 96.

9. For a number, greater than 10, to be Prime what may be the possible digit in the unit place? Number greater than 10 to be Prime should have 1, 3, 7 or 9 in unit place.

10. List all primes having 5 as the digit at its unit place. Only prime having 5 in its unit place is 5 itself.

11. What is the prime factorization of a number? Writing a number as product of only prime numbers is called the prime factorization of the number.

12. Give three pairs of prime numbers whose difference is 2. Two prime numbers whose difference is 2 are called twin primes Three pairs are: (3, 5); (5, 7) and (11, 13).

1. Write all the factors of a. 26 b. 32 a. Factors of 26 = 1, 2, 13, 26

I. Short Answer Type Questions

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b. Factors of 32 = 1, 2, 4, 8, 16 and 32

2. Write the lowest common multiple of 3 and 4.

Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, … Multiples of 4=4, 8, 12, 16, 20, 24,… Common multiples = 12, 24 Lowest common Multiple = 12.

3. Find the LCM of 15 and 20 For finding the LCM of 15 and 20, we use division method.

LCM of 15 and 20 = 5 X 3X 4 = 60.

4. Express each of the following as the sum of three odd prime numbers. a. 49 b. 35 a. 49 = 3 + 5 + 41 b. 35 = 5 + 7 + 23

5. Express each of the following numbers as the sum of two primes. a.36 b. 84 a. 36 = 17 + 19 b. 84 = 41 + 43

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6. Test the divisibility of the following numbers.

a. 4684 b. 723 c. 495 d. 2853 a. Given number is 4864. Hence, unit place is 4. So, it is divisible by 2. b. Sum of digits of the number = 7 + 2 + 3 =12 which is divisible by 3. Hence, number 723, is also divisible by 3. c. Here, unit place is 5. So, it is divisible by 5 and sum of digits = 4+9+5=18 So, it is divisible by 3. d. Sum of digits of the number = 2+8+5+3=18, which is divisible by 3 and 9 both.

7. Give the factorization of the following numbers. a. 136 b. 252 a. We have, 136

∴ 136 = 1 X 2 X 2 X 2 X 17 b. We have, 252

∴ 252 = 1 X 2 X 2 X 3 X 3X 7

8. Find the HCF of the following numbers using prime factorization method a. 84, 98 b. 72,108,180 a. For the HCF of 84 and 98

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84 = 2 X 2 X 3 X 7 ⇒ 98 = 2 X 7 X 7

HCF = 2 X 7 = 14

b. Do same as part (a)

HCF = 2 X 2 X 3 X 3 = 36.

9. Find the HCF of the following numbers, using division method. a. 58, 70 b. 24, 36, 40 a. For the HCF of 58, 70 ____ 58)70(1 58 12) 58(4 48 10)12(1 10 2)10(5 10 0 ∴ HCF of 58 and 70 is 2. b. First of all, find the HCF of 24 and 36. ____ 24) 36(1 24

12)24(2 24 0

∴ HCF of 12 and 40 is 4. Hence, HCF of 24, 36 and 40 is 4.

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10. Find the LCM of 112, 168, 266 by prime factorization method. For the LCM of 112, 168 and 266

112 = 2 X 2 X 2 X 2 X 7 = 24 x 7 168 = 2 x 2 X 2 X 3 X 7 = 23 X 3 X 7 266 = 2 X 7 X 19 ∴ LCM = 24 X 3 X 7 X 19 = 6384

11. Find the LCM of 18, 22, 40 by prime factorization division method. Do same as Q. 10 LCM = 2 X 9 X 11 X 20 = 3960.

12. In each of the following numbers, replace * by the smallest number to make it divisible by 3. a. 27 * 4 b. 53 * 46 For divisible by 3, the sum of the digits should be divisible by 3. a. Sum of digits of 27 * 4 = 2 + 7 + 4 = 13 ∴ replace * by 2 b. Sum of digits of 53 * 46 = 5 + 3 + 4 + 6 = 18 ∴ replace * by 0.

13. Give the prime factorization of 1224? We have, 1224

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∴ Prime factorization of 1224 is 2 X 2 X 2 X 3 X 3 X 17

14. Find the HCF of 513 and 783 For the HCF of 513 and 783, we use division method ____ 513) 783(1 -513 270) 513(1 -270___ 243)270(1 -243____ 27) 243(9 -243 X ∴ HCF of 513 and 783 is 27.

15. Find the LCM of 60 and 75. LCM of 60 and 75. = 3 X 5 X 4 X 5 = 300

16. Find the greatest number, which divides 753 and 1054 leaving remainder 3

and 4 respectively. The required number is the HCF of (753-3) and (1054-4) i.e. 750 and 1050.

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______ 750) 1050(1 -750 ____ 300)750(2 600 150) 300(2 300 0 ∴ The HCF of 750 and 1050 is 150 Hence, the required number is 150.

17. Determine the least number, which when divided by 3, 4 and 5 leaves remainder 2 in each case. First of all, we find the LCM of 3, 4 and 5.

LCM= 3 X 4 X 5 = 60 Hence, the required number = 60 + 2 = 62.

18. Three bells ring at intervals of 48, 60 and 90 min respectively. If all the three bells ring together at 10: 00 am. At what time will the three ball ring again that day?

LCM 2 X 3 X 2 X 5 X 2 X 2 X 3= 720 min In hour= 720 = 12 h 60 ∴ Bell ring together again = (10: 00) am + 12 h) = 10:00 pm

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19. Five bells begin to toll together at intervals of 9 s, 6 s, 4 s, 10 s and 58 s,

respectively, how many times will they toll together in the span of 1 h?

LCM = 2 X 2 X3 X 3 X 5 X 2= 360 In 1 h, the ring will toll together 3600 times = 10 times 360

1. Determine the least number which when divided by 3, 4 and 5 leaves remainder 2 in each case. LCM of 3, 4 and 5 = 2 X 2 X 3 X 5 = 60 Since, 60 is the least number exactly divisible by 3, 4 and 5.

To get the remainder 2, The least number = 60 + 2 = 62

2. The product of two numbers is 48. Their sum is 19. What are the numbers? Since the product of the required numbers is 48. So, the required numbers are factors of 48 such that their sum is 19. We first list all factors of 48. The factors are: 1 and 48; 2 and 24; 3 and 16:4 and 12; 6 and 8. Clearly, 3 and 16 are factors such that their sum is 19. Hence, the required numbers are 3 and 16.

II. Short Answer Type Questions

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3. Write all the numbers less than 100 which are common multiple of 3 and 4? Common multiples of 3 and 4 are multiples of 3 X 4, i.e., 12 12 X 1 = 12 12 X 2 = 24 12 X 3 = 36 12 X 4 = 48 12 X 5 = 60 12 X 6 = 72 12 X 7 = 84 12 X 8 = 96 ∴ Required numbers less than 100 are 12, 24, 36, 48, 60, 72, 84, and 96.

4. Determine the HCF of 216 and 1176. 216 X 2 X 2 X 2 X 3 X 3 X 3

and

Thus, 216 = 2 X 2X 2 X 3 X 3 X 3 1176 = 2 X 2 X 2 X 3 X 7 X7 Hence, HCF of 216 and 1176 = 24

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5. Reduce ퟐퟖퟗퟑퟗퟏ

to the lowest term

Thus, 289 = 1 X 17 X 17 And 391 =1 X 17 X 23 HCF = 1 X 17 = 17 Since the HCF of 289 and 319 = 17 Therefore,

= ÷

÷ =

6. Find the L. C. M of 40, 36 and 126 by prime factorization method.

Prime factorizations are: 40 = 2 X 2 X 2 X 5 = 2 3X 5 36 = 2 X 2 X 3 X 3 = 22 X 32

126 = 2 X 3 X 3 X7 = 2 X 32 X 7 ∴ LCM of 40, 36 and 126 = 23 X 32 X 5 X 7 = 8 X 9 X 5 X 7 = 2520.

7. Find the greatest number which divides 285 and 1249, leaving remainder 9 and 7. For greater number, we must find the HCF of (285-9) and (1249-7) i.e., HCF of 276 and 1242 _4_ 276) 1242( -1104

138)276(2 276

X Hence, the required number = 138.

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1. Write first five multiples of: i.5 ii. 8 iii. 9 i. Multiples of 5 = 5, 10, 15, 20, 25 ii. Multiples of 8= 8, 16, 2, 32, 40 iii. Multiples of 9 = 9, 18, 27, 36, 45

2. Write down separately the prime and composite numbers less than 20. Prime numbers and composite numbers less than 20 are: Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19 Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.

3. Find the common factors of: i. 4, 8 and 12 ii. 5, 15 and 25 i. Factors of 4 are 1, 2, 4 Factors of 8 are 1, 2, 4, 8 Factors of 12 are 1, 2, 3, 4, 6, 12 Thus, common factors of 4, 8 and 12 are 1, 2, 4. ii. Factors of 5 are 1, 5 Factors of 15 are 1, 3, 5, 15 Factors of 25 are 1, 5, 25 Thus, common factors of 5, 15 25 are 1, 5.

4. Find first three common multiples of: i. 6 and 8 ii. 12 and 18 i. Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72. Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72. Thus, first three common multiples are 24, 48, 72. ii. Multiples of 12 are 12, 24, 36,48, 60, 72, 84, 96, 108, 120. Multiples of 18 are 18, 36, 54, 72, 90, 108. Thus, first three common multiples are 36, 72, 108.

III. Short Answer Type Questions

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5. Give the prime factorization of 12, 650.

Therefore, 12,650 = 2 X 5 X 5 X 11 X 23 = 2 X 52 X 11 X 23

6. Find the greatest number which divides 2, 011 and 2, 623, leaving remainders 9 and 5 respectively. We must find the greatest number which divides (2,011-9) = 2, 002 and (2, 623-5) = 2, 618 exactly. So, the required number = HCF of 2, 002 and 2, 618. __1_ 2002 )2618 -2002____ 616)2002(4 -1846____ 154) 616(4 -616 0 So, the required number is 154.

7. State whether the following statements are True or False: i. Every multiple of a number is greater than or equal to the number. ii. Sum of two consecutive odd numbers is always divisible by 4. iii. If a number is divisible by both 2 and 3, then it is divisible by 12. iv. If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 9. 1. True 2. True 3. False 4. False

8. Find the LCM of 160, 170 and 90. We have,

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Here, LCM of given numbers = 2 X 5 X 9 X 16 X 17 = 24, 480.

9. What are co-primes? Give examples of five pairs of co-primes. Are co-primes always Prime? If no, illustrate your answer by an example. Two numbers are said to be co-prime if they do not have a common factor other than 1. Examples of co-prime are 2, 3; 3, 4; 5, 6; 8, 13 etc. No, both the numbers of co-primes need not be Prime. For example, 14, 15 are co-primes, while none of 14 and 15 is a prime number.

10. Find all the prime factors of 1, 729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors. Prime factors of 1, 729 are:

So, prime factors of 1 ,729 are 7, 13, 19 Arranging them in ascending order, we get 7, 13, 19 Here, each factor is 6 more than the previous factor, i. e.,

7 + 6 = 13 13 + 6 = 19

So, relation between two consecutive prime factors is that the difference between them is 6.

11. Write the largest 4-digit number and give its prime factorization. The largest 4-digit number is 9999 so far its prime factorization we use division method.

∴ 9999 = 3 X 3X 11 X 101

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12. Fill in the blanks. i. If a number has ____________ in ones place, then it is divisible by 10. ii. The HCf of two or more given numbers is the highest of their common ____. iii. Two number having only 1 as a common factor are called ________ numbers. iv. If the sum of the digits in a number is a _______of 3, then the number is divisible by 3. v. A number for which the sum of all its factors is equal to twice the number is called a _______________ number. vi. If the difference between the sum of digits at odd places (from the right) and the sum of digits at even places (from the right) of a number is either 0 or divisible by __________, then the number is divisible by 11. i. 0 ii. Factors iii. Co-prime iv. Multiple v. Perfect vi. 11

1. Find the greatest number that will divide 445, 572 and 699, leaving remainders 4, 5 and 6 respectively. For the required number, we must find the HCF of (445-4), (572-5) and (699-6) i.e. 441, 567 and 693. ∴ _______ 441)567(1 441 126)441(3 378____ 63) 126(2 126 0 ∴ HCF of 441 and 567 is 63. Now, we will find the HCF of 63 and 693 HCF of 63 and 693 is 63. Hence, HCF of 441, 567 and 693 = 63 ∴ The required number = 63.

I. Long Answer Type Questions

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2. A merchant has 120 L of oil of one kind, 180 L of another kind and 240 L

of a third kind. He wants to sell the oil by filling the three kinds of oil in this of equal capacity. What should be the greatest capacity of such tin? Given, merchant have 120 L of oil one kind, 180 L of another kind and 240 L of a third kind. The required capacity of a tin will be HCF of 120, 180 and 240.

______ 120) 180(1 120 60)120(2 120 0

HCF of 120 and 180 is 60. Now, we find the HCF of 60 and 240 ____ 60) 240(4 240 0 The HCF of 60 and 240 is60. Hence, HCF 120, 180 and 240 is 60. ∴ The required capacity of a tin = 60 L

3. Fatima wants to mail three parcels to three village schools. She find that the postal charges are ₹ 20, ₹ 28 and ₹ 36, respectively. If she wants to buy stamps only of the denomination, what is the greatest denomination of stamps she must buy to mail the three parcels? The given postal charges for three parcels are ₹ 20, ₹ 28 and ₹ 36. The greatest denomination of stamps is the HCF of 20, 28 and 36. ____ 20) 28(1 20 8)20(2 16 4)20(5 20 0 HCF of 20 and 28 is 4. Now, we find the HCF of 4 and 36. HCF of 4 and 36 is 4.

Hence, HCF of 20, 28 and 36 is 4. ∴ Fatima would by stamp of denomination of ₹ 4.

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4. Three boys step off together from the same place. If their steps measure

36 cm, 48 cm and 54 cm, at what distance from the starting point will they again step together? Given, their steps measure is 36 cm, 48 cm and 54 cm. We must find the LCM of 36, 48, 54.

LCM of 36, 48 and 54 = 2 X 2 X 3 X 3 X 4 X 3 = 432. They again step together from the starting point at 432 cm = 4 m 32 cm.

5. Three brands A, B and C of biscuits are available in packets of 12, 15 and 21 biscuits, respectively. If a a shopkeeper wants to buy an equal number of biscuits of each brand, what is the minimum number of packets of each brand, he should buy? In brand A, number of biscuits = 12 In brand B, number of biscuits = 15 In brand C, number of biscuits = 21 First of all, we find the LCM of 12, 15 and 21

LCM of 12, 15 and 21 = 3 X 4 X 5 X 7 = 420

Now, number of pockets of brand A = ퟒퟐퟎퟏퟐ

= 35

Number of packets of brand B = ퟒퟐퟎퟏퟓ

= 28

Number of packets of brand C = ퟒퟐퟎퟐퟏ

= 20

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6. On a morning walk, three persons step off together and their and their

steps measure 40 cm, 42 cm and 46 cm, respectively. a. What is the minimum distance each should walk so that each can cover the same distance in complete steps? b. What are the benefits of morning walk? a. The steps measure of each person is 40 cm 42 cm and 46 cm. So, the minimum distance each should walk is the LCM of 40, 42 and 46.

LCM of 40, 42 and 46 = 2 X 20 X 21 X 23 = 19320 cm = 193.2m b. Morning walk benefits us in many ways. It maintains our good health. Some benefits of morning walks are listed below: i. Strengthen our heart ii. Delays or prevents major diseases or illness. iii. Reduce blood pressure and the risk of stroke iv. Reduces cholesterol v. Boost immune system

7. A merchant has 130 L of oil of one kind, 190 L of another kind and 250 L of a third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin? The greatest capacity of the required measure will be equal to the HCF of 130, 190 and 250L prime factorization of 130, 190 and 250.

130 = 2 X 5 X 13 190 = 2 X 5 X 19 250 = 2 X 5 X 5 X 5 Common factors of 130, 190 and 250 = 2X 5= 10 Hence, greatest capacity of tin = 10L

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8. Find a 4-digit odd number using each of digits 1, 2, 4 and 5 only once such

that when the first and last digits are interchanged, it is divisible by 4. The 4- digit number will be an odd number, if the unit place digit is an odd number (i.e. 1 or 5). Total such odd numbers are, 4125, 4215, 1245, 2145, 2415, 4251, 4521, 5241, 5421, 2451, 2541. Also we know that, any 4-digit numbers is divisible by 4, if the last two digit number is divisible by 4. Consider a number 4521, if we interchange the first and the last digit, the new number will be 1524. Here, we see that the last two digit (i.e. 24) is divisible by 4. So, the number 1524 is divisible by 4. Hence, the required 4-digit number is 4521.

9. Monica, Heronica and Rajat begin to jog around a circular stadium. The complete their revolution in 42 s, 56 s, and 63 s respectively. How many seconds after will they be together at the starting point?

LCM = 2X 3 X 7 X 2 X 2 X 3 = 504 s Hence, after 504 s, they will be together at starting point.

10. Find the greatest possible length, which can be used to measure exactly the length 7m, 3m 85 cm and 12 m 95 cm.

____ 385) 700(1 385 315) 385(4 315___

70)315(4 280___ 35) 70(2 70 0

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_____

35) 1295(37 -105 245 245 0 Hence, required length is 37 cm.

1. A box contains 5 strips having 12 capsules of 500 mg medicine in each capsule. Find the total weight in grams of medicine in 32 such boxes. Given, in each strip there are 12 capsules. Also, given weight of one capsule is 500 mg ∴ Weight of 12 capsules = 12 X 500 = 6000 mg ∴ Weight of 1 strip = weight of 12 capsule = 6000 mg ∴ Weight of 5 strips = 5X weight of one strip = 5 X 6000 = 30, 000 mg ∴ Weight of 1 box = weight of 5 strips = 30,000 mg ∴ Weight of 32 boxes= 32 X weight of 1 box = 32 X 30,000 mg

= gm

= 960 gm

[∴ 1 gm = 1000 mg and 1 mg = gl

Hence, the total weight of 32 medicine box is 960 gm.

2. The floor of a room is 8 m 96 cm long and 6 m 72 cm broad. Find the minimum number of square tiles of the same size needed to cover the entire floor. Given, length of the floor = 8m 96cm

II. Long Answer Type Questions

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= 8X 100cm + 96 cm [∴ 1m = 100cm] = (800 + 96) cm = 896 cm And breadth of the floor = 6m 72cm = 6X 100 cm + 72 cm [∴ 1m = 100 cm] = 672 cm Now, size of the square tile = HCF of 896 and 672 prime factorization of 896 and 672

896 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 7 672 = 2 X 2 X 2 X 2 X 2 X 2 X 7 Common factors of 896 and 672 = 2 X 2 X 2 X 2 X 2 X 7 = 224 ∴ Minimum no. of square tiles = Area of floor Area of square tile

=

[∴ area of square = (side)2] =

= 4X 3 = 12

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3. A Merchant has 120 liters of oil of one kind, 180 liters of another kind and

240 liters of a third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

The greatest capacity of the required measure will be equal to the HCF of 120, 180 and 240 L. Prime factorization of 120, 180 and 240.

120 = 2 X 2 X 2 X 3 X 5 180 = 2 X 2 X 3 X 3 X 5 240 = 2 X 2 X 2 X 2 X 3 X 5 Common factors of 120, 180 and 240 = 2 X 2 X 3 X 5 = 60 Hence, greatest capacity of tin = 60 L.

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1. Write all the factors of the following numbers: i. 24 ii. 15 iii. 21 iv. 27 v. 12 vi. 20 vii. 18 viii. 23 ix. 36 i. 24 Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. ii. 15 Factors of 15 are 1, 3, 5, 15 iii. 21 Factors of 21 are 1, 3, 7, 21. iv. 27 Factors of 27 are 1, 3, 9, 27. v. 12 Factors of 12 are 1, 3, 4, 6, 12 vi. 20 Factors of 20 are 1, 2, 4, 5, 10, 20. vii. 18 Factors of 18 are 1, 2, 3, 6, 9, 18 viii. 23 Factors of 23 are 1, 23 ix. 36 Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36

III. Long Answer Type Questions

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2. Express each of the following numbers as the sum of three odd primes:

i. 21 ii. 31 iii. 53 iv. 61 i. 21 It can be written as 21 = 3 + 5 + 13 Here, 3, 5 and 13 are three odd primes. ii. 31 It can be written as 31 = 3 + 5 + 23 Here, 3, 5 and 23 are three odd primes. iii. 53 It can be written as 53 = 13 + 17 + 23 Here, 13, 17 and 23 are three odd primes. iv. 61 It can be written as 61 = 7 + 13 + 41 Here, 7, 13 and 41 are three odd primes.

3. Using divisibility tests, determine which of the following number are divisible by 11: i. 5, 445 ii. 10, 824 iii. 71, 38, 965 iv. 7, 01, 69, 308 v. 1, 00, 00, 001 vi. 9, 01, 153 i. 5, 445 Sum of the digits in odd places = 5 + 4 = 9 Sum of the digits in even places = 5 + 4 = 9 Difference of two sums = 9 – 9 = 0 So 5, 445 is divisible by 11. ii. 10, 824 Sum of digits in odd places = 8 + 4 + 1 = 13 Sum of digits in even place = 0 + 2 = 2 Difference of two sums = 13 – 2 = 11 which is divisible by 11

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Therefore, 10, 824 is divisible by 11. iii. 71, 38, 965 Sum of digits in odd places = 7 + 3 + 9 + 5 = 24 Sum of digits in even places = 1 + 8 + = 15 Difference of two sums = 24- 15 = 9, which is not divisible by 11 Therefore, 71, 38, 965 is not divisible by 11. iv. 7, 01, 69, 308 Sum of digits in odd places = 7 + 1 + 9 + 0 = 17 Sum of digits in even places = 0+ 6 + 3 + 8 = 17 Difference of two sums = 17 – 17 = 0 Therefore, 7, 01, 69, 308 is divisible by 11. V. 1, 00, 00, 001 Sum of digits in odd places = 1 Sum of digits in even places = 1 Difference of two sums = 1 – 1 = 0 Hence, it is divisible by 11. vi. 9, 01, 153 Sum of digits in odd places = 9 + 1 + 5 = 15 Sum of digits in even places = 0 + 1 + 3 = 4 Difference of two sums 1 – 1 = 0 Hence, it is divisible by 11. vii. 9, 01, 153 Sum of digits in odd places = 9 + 1 + 5 = 15 Sum of digits in even places = 0 + 1 + 3 = 4 Difference of two sums = 15 – 4 = 11 Hence, it is divisible by 11.

4. Test the divisibility of the following numbers by 9: i. 2, 358 ii. 3, 333 iii. 9, 8712 iv. 2, 57, 106 v. 6, 47, 514 vi. 3, 26, 999 i. 2, 358 Sum of digits = 2 + 3 + 5 + 8 = 18, which is divisible by 9 So, the given number is divisible by 9.

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ii. 3, 333 Sum of digits = 3 + 3 + 3 + 3 = 12, which is not divisible by 9 So, the given number is not divisible by 9. iii. 98, 712 Sum of digits = 9 + 8 + 7 + 1 + 2 = 27, which is divisible by 9. So, the number is divisible by 9. iv. 2, 57, 106 Sum of digits = 2 + 5 + 7 + 1 + 0 + 6 = 21 So, the given number is not divisible by 9. v. 6, 47, 514 Sum of digits = 6 + 4 + 7 + 5 + 1 + 4 = 27, which is divisible by 9 So, the given number is divisible by 9. vi. 3, 26, 999 Sum of digits = 3 + 2 + 6 + 9 + 9 + 9 = 38 So, the given number is not divisible by 9.

5. Match the following

i. 4 a. Multiple of 16 ii. 5 b. Multiple of 81 iii. 11, 96, 625 c. Multiple of 25 iv. 34, 307 d. Factor of 16 v. 6, 790 e. Factor of 81 vi. 9 f. Factor of 25 vii. 64 g. Divisible by 7 viii. 75 h. Divisible by 10 i. Divisible by 11

i. d ii. f iii. i iv. g v. h vi. e vii. a viii. c

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6. Find the HCF of the following numbers:

i. 18, 48 ii. 34, 102 iii. 70, 105, 175 iv. 91, 112, 49 v. 18, 54, 81

i. 18, 48

We have

So, 18 = 2 X 3 X 3

48 = 2 X 2 X 2 X 2 X 3

Hence, HCF of 18 and 48 = 2 X 3 = 6

ii. 34, 102

We have,

34 = 2 X 17

102 = 2 X 3 X 17

Hence, HCF of 34 and 102 = 2 X 17 = 34

iii. 70, 105, 175

We have,

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So, 70 = 2 X 5 X 7

105 = 3 X 5 X 7

175 = 5X 5 X 7

Hence, HCF of 70, 105 and 175 = 5 X 7 = 35

iv. 91, 112, 49

We have,

91 = 7 X 13

112 = 2 X 2 X 2 X 2 X 7

49 = 7 X 7

Hence, HCF of 91, 112 and 49 = 7

v. 18, 54, 81

We have,

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So, 18 = 2 X 3 X 3

54 = 2 X 3 X 3 X 3

81 = 3 X 3 X 3 X 3

Hence, HCF of 18, 54 and 81 = 3 X = 9.

7. Find the LCM of the following numbers:

i. 9 and 4 ii. 12 and 5 iii. 6 and 5 iv. 15 and 4

Observe a common property in the obtained LCMs. Is LCM the product of two numbers in each case?

i. 9 and 4

We have,

So, 9 = 3 X 3

4 = 2 X 2

Therefore, LCM of 9 and 4 = 2 X 2 X 3 X 3

= 4 X 9 = 36.

ii. 12 and 5

We have,

So, 12 = 2 X 2 X 3

5 = 5

Therefore, LCM of 12 and 5 = 2 X 2 X 3 X 5 = 12 X 5 = 60

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iii. 6 and 5 We have,

So, 6 = 2 X 3

5 = 5

Therefore, LCM of 6 and 5 = 2 X 3 X 5 = 6 X 5 = 30.

iv. 15 and 4

We have,

So, 15 = 3 X 5

4 = 2 X 2

Therefore, LCM of 15 and 4 = 2 X 2 X 3X 5 = 4 X 15 = 60

The common property observed is, in each case LCM is a multiple of 3.

Yes, in each case LCM is equal to product to product of two numbers.

8. Find the least number which when divided by 20, 25, 30 and 36 leaves a remainder 4 in each case.

We first find the LCM of 20, 25, 30, 36.

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Thus, LCM = 2 X 2 X 3 X 3 X 5 X 5

= 4 X 9 X 25 = 36 X 25

= 900.

900 is the least number, which when divided by give numbers leaves remainder 0. But we need least number which leaves remainder 4 in each case.

Therefore, the required number is 4 more than 900.

The required least number = 900 + 4 = 904.

9. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.

The maximum capacity of such a container will be HCF of 403, 434 and 465 and container has to measure the tankers in a way that the count is exact. So, we have,

Hence, 403 = 13 X 31

434 = 2 X 7 X 31

465 = 3 X 5 X 31

So, HCF of 403, 434 and 465 = 31

Therefore, the maximum capacity of the container is 31 litres.

10. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 am, at what time will they change simultaneously again?

The light changes simultaneously at 7 am so to find the time at which they will change Hence, required time = LCM of 48, 72, 108.

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Hence, LCM of 48, 72, 108 = 2 X 2 X 2 X 2 X 3 X 3 X 3

= 4 X 4 X 3 X 9 = 12 X 36 = 432

So, all the traffic lights will change simultaneously after 432 seconds.

432 seconds = 7 minutes 12 seconds [7 min = 420 seconds)

So, traffic lights will change after 7 minutes 12 seconds.

11. Find the HCF and LCM of 861, 1, 353.

We first find HCF of two numbers

Therefore, 861 and 1,353 = 3 X 41 = 123

Now, we have to find LCM of these two numbers.

So, we know, 퐿퐶푀 = ( ) ( )

LCM = ,

X 7 X 1, 353 = 9, 471

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1. i. Find the smallest number which when diminished by 3 is divisible by 21, 28,

36 and 45.

ii. If one number is 8 and other number is 12, LCM of 8 and 12 = 24, HCF of 8

and 12= 4

Then verify, product of two number = LCM X HCF

i. For smallest number, we must find the LCM of 21, 28, 36 and 45.

LCM of 21, 28, 36, 45 = 2 X 2 X 3 X 3 X 5 X 7

= 1260

Hence required number = (1260 + 3) = 1263.

ii. Since, one number = 8

And other number = 12, than

LHS = Product of two numbers = 8 X 12 = 96 RHS = LCM X HCF = 24 X 4 = 96 Hence, LHS = RHS

I. High Order Thinking Skills Questions

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1. Find a 4- digit odd number using each of the digits 3, 2, 6and 5 only once such that when the first and the last digits are interchanged, it is divisible by 4.

To make a number divisible by 4, the last 2-digits of the number must be divisible by 4.

The number is an odd number, so it must have either 3 or 5 in its units place, which on reversing with the first digit make the number divisible by 4. So, the number is 6, 235.

On reversing the first and last digits, we get 5236, which is divisible by 4.

2. Sam wants to stack his three sets of English, Hindi and Science books in a way that all the books are stored topic-wise and the height of each stack is the same. The number of English books is 105, the number of Hindi books is 140 and the number of science books in 175. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Science books.

To arrange the books as required, we have to find the largest number that divides 105, 140 and 175 exactly.

Such a number is their HCF.

The HCF of 105, 140 and 175 is

Thus 105 = 3 X 5 X 7

140 = 2 X 2 X 5 X 7

175 = 5 X 5 X 7

So, the HCF of 105, 140 and 175 is 5 X 7 = 35.

Hence, there must be 35 books in each stack.

Now, No. of stacks of English books = . .

= = 3

II. High Order Thinking Skills Questions

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Now, No. of stacks of Hindi books = . .

= = 4

Now, No. of stacks of Science books = . .

= = 5

3. In a colony of 100 blocks of flats numbering 1 to 100, a school van stops at every sixth block while a school bus stops at every tenth block. On which stops will both of them stop if they start from the entrance of the colony?

The common stop at which both van and bus stop is the common multiple of 6 and 10.

Multiples of 6 less than 100 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96.

Multiples of 10 less than 100 are 10, 20, 30, 40, 50, 60, 70, 80, 90

Common multiples of 6 and 10 are 30, 60 and 90

So, the school van and bus both will stop together at blocks 30, 60 and 90.

4. The circumference of four wheels are 50 cm, 60 cm, 75 cm, and 100 cm respectively. They start moving simultaneously. What least distance should they cover so that each wheel makes complete number of revolutions?

The circumference of four wheel = 50 cm, 60 cm, 75cm and 100 cm.

The least distance they should cover so that each wheel makes complete number of revolution is the LCM of the circumference of four wheels.

The LCM of 50 cm, 60cm, 75 cm and 100 cm is

LCM of 50, 60, 75 and 100 is 2 X 2 X 3 X 5 X 5 = 300

So, the least distance the wheel should cover so that each wheel makes complete number of revolution is 300 cm or 3 m.

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1. i. Find the HCF of 513 and 783 using long division (Algorithm) method.

ii. Test the divisibility of 83745 by 9.

i. Since, dividend = 783

and divisor = 513

Then

_____ 513) 783(1 -513 270)-270(1 243(270(1 -243 27) 243 X Hence, HCF of 783 and 513 = 27 ii. Since a number is divisible by 9, if the sum of its digits is divisible by 9. Therefore, sum of digits of given no. = 8 + 3 + 7 + 4 + 5 = 27, which is divisible by 9. 2. i. In a shop, there are 3 clocks which chime at intervals of 15, 20 and 30 minutes respectively. They all chime together at 10 a.m. At what time will they all chime together again? Required time = LCM of 15, 20, 30 minute

I. Value Based Questions

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LCM of 15, 20, 30 = 2 X 2 X 3 X 5 = 60 Therefore, all the clocks will chime together again After 60 minutes, i.e., 1 hour, i.e., at 11 a.m. ii. Write the smallest prime and composite number. Smallest prime number = 2 And smallest composite number = 4.

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Mathematics Chapter 3: Playing With Numbers - Pinkz Public