Class 6 Maths Chapter 5 Solutions: Prime Time

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Solution sheet 1

Page no. 108 · Figure it out

NCERT Class 6 Maths Chapter 5, question 1 — Source question

1. At what number is ‘idli-vada’ said for the 10th time?

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150.

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NCERT Class 6 Maths Chapter 5, question 2 — Source question

2. If the game is played for the numbers from 1 till 90, find out: a. How many times would the children say ‘idli’ (including the times they say ‘idli- vada’)?

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30 times. b. How many times would the children say ‘vada’ (including the times they say ‘idli- vada’)? Ans. 18 times. c. How many times would the children say ‘idli-vada’? Ans. 6 times.

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NCERT Class 6 Maths Chapter 5, question 3 — Source question

3. What if the game was played till 900? How would your answers change?

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If the game was played till 900, idli-vada would be said 60 times by the children. ‘Vada’ will be said 180 time and ‘idli’ will be said 300 times.

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NCERT Class 6 Maths Chapter 5, question 4 — Source question

4. Is this figure somehow related to the ‘idli-vada’ game? Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.

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Yes, the common numbers represent the numbers when to say ‘idli-vada’. 1 Page no. 109 Which of the following could be the other number: 2, 3, 5, 8, 10? Ans. The other number will be 8. Page no. 110

Source worked solution for Class 6 Chapter 5, question 4, part 1 — Source worked answer · part 1

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Solution sheet 2

Page no. 109 · Page no. 110

NCERT Class 6 Maths Chapter 5, question 5 — Source question

What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all.

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Jump size of 3 or 5 will take us to 15 & 30. Other possible jump sizes are 1, 15.

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Page no. 110

NCERT Class 6 Maths Chapter 5, question 6 — Source question

1. Is there anything common among the shaded numbers?

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All shaded numbers are multiples of 3.

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Page no. 110

NCERT Class 6 Maths Chapter 5, question 7 — Source question

2. Is there anything common among the circled numbers?

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All circled numbers are multiples of 4.

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NCERT Class 6 Maths Chapter 5, question 8 — Source question

3. Which numbers are both shaded and circled? What are these numbers called?

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36, 48, 60. These numbers are called common multiples of 3 and 4. Section 5.1 Page no. 110 Figure it out

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NCERT Class 6 Maths Chapter 5, question 9 — Source question

1. Find all multiples of 40 that lie between 310 and 410.

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Multiples of 40 that lie between 310 and 410 are: 320, 360, 400.

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Page no. 110 · Figure it out

NCERT Class 6 Maths Chapter 5, question 10 — Source question

2. Who am I? a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8. b. I am a number less than 100. Two of my factors are 3 and 5.One of my digits is 1 more than the other.

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a. 35 b. 45 2

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Solution sheet 3

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NCERT Class 6 Maths Chapter 5, question 11 — Source question

3. A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.

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6.

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NCERT Class 6 Maths Chapter 5, question 12 — Source question

4. Find the common factors of: a. 20 and 28 b. 35 and 50 c. 4, 8 and 12 d. 5, 15 and 25

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a. Common factors of 20 and 28 = 1, 2, 4. b. Common factors of 35 and 50 = 1, 5. c. Common factors of 4, 8 and 12 = 1, 2, 4. d. Common factors of 5, 15 and 25 = 1, 5

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NCERT Class 6 Maths Chapter 5, question 13 — Source question

5. Find any three numbers that are multiples of 25 but not multiples of 50.

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Some such numbers are: 25, 75, 125, 175.

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NCERT Class 6 Maths Chapter 5, question 14 — Source question

6. Anshu and his friends play the ‘idli-vada’ game with two numbers, which are both smaller than 10. The first time anybody says ‘idlivada’ is after the number 50. What could the two numbers be which are assigned ‘idli’ and ‘vada’?

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7, 8 8, 9 (Try for other possibility)

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NCERT Class 6 Maths Chapter 5, question 15 — Source question

7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?

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The jump sizes are 1, 2, 7 or 14

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NCERT Class 6 Maths Chapter 5, question 16 — Source question

8. In the diagram below, Guna has erased all the numbers except the common multiples. Find out what those numbers could be and fill in the missing numbers in the empty regions.

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Multiples of 3 Multiples of 4 Think of some more.

Source worked solution for Class 6 Chapter 5, question 16, part 1 — Source worked answer · part 1

Think of some more.

NCERT Class 6 Maths Chapter 5, question 17 — Source question

9. Find the smallest number that is a multiple of all the numbers from 1 to 10 except for 7.

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360

Source worked solution for Class 6 Chapter 5, question 17, part 1 — Source worked answer · part 1

Think of some more.

NCERT Class 6 Maths Chapter 5, question 18 — Source question

10. Find the smallest number that is a multiple of all the number from 1 to 10.

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2520 3 Section 5.2 Page No. 113

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Solution sheet 4

Section 5.2 · Page No. 113

NCERT Class 6 Maths Chapter 5, question 19 — Source question

How many prime numbers are there from 21 to 30? How many composite numbers are there from 21 to 30?

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Prime numbers from 21 to 30: 2 (23,29) Composite numbers from 21 to 30: 8 Page No. 114 Figure it out

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NCERT Class 6 Maths Chapter 5, question 20 — Source question

1. We see that 2 is a prime and also an even number. Is there any other even prime?

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No.

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Page No. 114 · Figure it out

NCERT Class 6 Maths Chapter 5, question 21 — Source question

2. Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?

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The smallest difference between two successive primes is one (3-2=1) the largest difference is 8 (97-89).

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NCERT Class 6 Maths Chapter 5, question 22 — Source question

3. Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes?

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No. The least number of primes occur in the decades 91 to 100 and most number of primes and occur in the decades 1 to 10, 11 to 20.

Source worked solution for Class 6 Chapter 5, question 22, part 1 — Source worked answer · part 1

previous page? Which decades have the least number of primes? Which have the

NCERT Class 6 Maths Chapter 5, question 23 — Source question

4. Which of the following numbers are prime: 23, 51, 37, 26?

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23 and 37.

Source worked solution for Class 6 Chapter 5, question 23, part 1 — Source worked answer · part 1

previous page? Which decades have the least number of primes? Which have the

NCERT Class 6 Maths Chapter 5, question 24 — Source question

5. Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

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Pairs of prime numbers less than 20 whose sum is a multiple of 5 are (2,3), (3,7), (2,13) (Try for other possibilities).

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NCERT Class 6 Maths Chapter 5, question 25 — Source question

6. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.

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17 and 71, 37 and 73, 79 and 97.

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NCERT Class 6 Maths Chapter 5, question 26 — Source question

7. Find seven consecutive composite numbers between 1 and 100.

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90, 91, 92, 93, 94, 95, 96. 4

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Solution sheet 5

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NCERT Class 6 Maths Chapter 5, question 27 — Source question

8. Twin primes are pairs of primes having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin primes between 1 and 100.

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3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and 73.

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NCERT Class 6 Maths Chapter 5, question 28 — Source question

9 Identify whether each statement is true or false. Explain. a. There is no prime number whose units digit is 4. b. A product of primes can also be prime. c. Prime numbers do not have any factors. d. All even numbers are composite numbers. e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.

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a. True, unit digit 4 means even number. We know that 2 is the only even prime number. b. False, product of primes becomes a composite number. c. False, prime numbers have exactly two factors: 1 and the number itself. d. False, all even numbers except 2 are composite numbers e. True, as every pair, other than (2, 3), contains an even number which is a composite number.

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NCERT Class 6 Maths Chapter 5, question 29 — Source question

10. Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

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105 = 3 × 5 × 7 is the only number which is the product of three distinct prime numbers.

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NCERT Class 6 Maths Chapter 5, question 30 — Source question

11. How many three-digit prime numbers can you make using each of 2, 4 and 5 once?

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None

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NCERT Class 6 Maths Chapter 5, question 31 — Source question

12. Observe that 3 is a prime number, and 2 × 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.

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Yes, 11 = 2 × 5 + 1, 47 = 2 × 23 + 1, 83 = 2 × 41 + 1 23 = 2 × 11 +1, 59 = 2 × 29 +1 (Try other possibilities.) 5 Section 5.3 Page No. 115 Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures? Check if these pairs are safe: a. 15 and 39 b. 4 and 15 c. 18 and 29 d. 20 and 55 Ans. Safe pairs are: 4 and 15, 18 and 29. Page no. 116

Source worked solution for Class 6 Chapter 5, question 31, part 1 — Source worked answer · part 1

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Solution sheet 6

Page no. 116

NCERT Class 6 Maths Chapter 5, question 32 — Source question

Which of the following pairs of numbers are co-prime? a. 18 and 35 b. 15 and 37 c. 30 and 415 d. 17 and 69 e. 81 and 18

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The pairs of co-prime numbers are: a. 18 and 35 b. 15 and 37 d. 17 and 69

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NCERT Class 6 Maths Chapter 5, question 33 — Source question

While playing the ‘idli-vada’ game with different number pairs, Anshu observed something interesting! 1. Sometimes the first common multiple was the same as the product of the two numbers. 2. At other times the first common multiple was less than the product of the two numbers. Find examples for each of the above. How is it related to the number pair being co- prime?

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1. Examples when the first common multiple is the product of the two numbers: 3, 5; 3, 7 and 4, 9. 2. Examples when the first common multiple is less than the product of the two numbers: 3, 6; 3, 12 and 6, 15 Whenever the number pair is co-prime the first common multiple is the product of the two numbers. 6 Section 5.4 Page no. 120 Figure it out

Source worked solution for Class 6 Chapter 5, question 33, part 1 — Source worked answer · part 1

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Page no. 120 · Figure it out

NCERT Class 6 Maths Chapter 5, question 34 — Source question

1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

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64 = 2 × 2 × 2 × 2 × 2 × 2 104 = 2 × 2 × 2 × 13 105 = 3 × 5 × 7 243 = 3 ×3 ×3 ×3 ×3 320 = 2 × 2 × 2 × 2 × 2 × 2 × 5 141 = 3 × 47 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 729 = 3 ×3 ×3 × 3 ×3 ×3 1024 = 2 × 2 × 2 × 2 × 2 ×2 × 2 × 2 × 2 × 2 1331 = 11 × 11 × 11 1000 = 2 × 2 × 2 × 5 × 5 × 5

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NCERT Class 6 Maths Chapter 5, question 35 — Source question

2. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?

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198 = 2 × 3 × 3 × 11

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NCERT Class 6 Maths Chapter 5, question 36 — Source question

3. Find three prime numbers, all less than 30, whose product is 1955.

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1955 = 5 × 17 × 23

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NCERT Class 6 Maths Chapter 5, question 37 — Source question

4. Find the prime factorisation of these numbers without multiplying first a. 56 × 25 b. 108 × 75 c. 1000 × 81

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a. 56 × 25 = 2 × 2 × 2 × 7 × 5 × 5 b. 108 × 75 = 2 × 2 × 3 × 3 × 3 ×3 × 5 × 5 c. 1000 × 81 = 2 × 2 × 2 × 3 × 3 × 3 ×3 × 5 × 5 × 5

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NCERT Class 6 Maths Chapter 5, question 38 — Source question

5. What is the smallest number whose prime factorisation has: a. three different prime numbers? b. four different prime numbers?

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7 a. 2 × 3 × 5 = 30 b. 2 × 3 × 5 × 7 = 210 Page no. 122 Figure it out

Source worked solution for Class 6 Chapter 5, question 38, part 1 — Source worked answer · part 1

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Page no. 122 · Figure it out

NCERT Class 6 Maths Chapter 5, question 39 — Source question

1. Are the following pairs of numbers co-prime? Guess first and then use prime

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factorisation to verify your a. 30 and 45 b. 57 and 85 c. 121 and 1331 d. 343 and 216 Ans. a. No 30 = 2 × 3 × 5 45 = 3 × 3 × 5 b. Yes 57 = 19 × 3 85 = 17 × 5 c. No 121 = 11 × 11 1331 = 11 × 11 × 11 c. Yes 343 = 7 × 7 × 7 216 = 2 × 2 × 2 × 3 × 3 × 3.

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NCERT Class 6 Maths Chapter 5, question 40 — Source question

2. Is the first number divisible by the second? Use prime factorisation. a. 225 and 27 b. 96 and 24 c. 343 and 17 d. 999 and 99 3 × 3 × 5 × 5

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a. No ( 3 × 3 × 3 ) 2 × 2 × 2 × 2 × 2 × 3 b. Yes ( 2 × 2 ×2 ×3 ) 7 × 7 × 7 c. No ( 17 ) 3 × 3 × 3 × 3 × 37 d. No ( 3 × 3 × 11 ) 8

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Solution sheet 9

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NCERT Class 6 Maths Chapter 5, question 41 — Source question

3. The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? Does one of them divide the other?

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No, they are not co-prime. No, one of them is not dividing the other.

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NCERT Class 6 Maths Chapter 5, question 42 — Source question

4. Guna says, “Any two prime numbers are co-prime”. Is he right?

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Yes, for example 2, 3; 3,11 (Try other examples) Section 5.5 Page no. 124

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Section 5.5 · Page no. 124

NCERT Class 6 Maths Chapter 5, question 43 — Source question

Is 8536 divisible by 4?

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Yes, as 36 is divisible by 4.

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Section 5.5 · Page no. 124

NCERT Class 6 Maths Chapter 5, question 44 — Source question

Consider these statements: 1. Only the last two digits matter when deciding if a given number is divisible by 4. 2. If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4. 3. If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4. Do you agree? Why or why not?

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1. Yes 2. Yes 3. Yes Yes,we agree. Consider numbers 124,364,4028 etc. These are divisible by 4.Check for more numbers. Page no. 125

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Page no. 125

NCERT Class 6 Maths Chapter 5, question 45 — Source question

Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe?

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The numbers between 120 & 140 divisible by 8 are = 128, 136. The numbers between 1120 & 1140 divisible by 8 are = 1128, 1136. The numbers between 3120 & 3140 divisible by 8 are = 3128, 3136. 9 if the number formed by one’s, ten’s and hundred’s digit is divisible by 8, then the number is divisible by 8.

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Solution sheet 10

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NCERT Class 6 Maths Chapter 5, question 46 — Source question

Change the last two digits of 8560 so that the resulting number is a multiple of 8.

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8552 is a multiple of 8.

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NCERT Class 6 Maths Chapter 5, question 47 — Source question

Consider this statement: 1. Only the last three digits matter when deciding if a given number is divisible by 8. 2. If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8. 3. If the original number is divisible by 8, then the number formed by the last three digits is divisible by 8. Do you agree? Why or why not?

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a. Yes b. Yes c. Yes Yes. Some examples are: 8576,7648,5024. Try some more. Figure it out

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Figure it out

NCERT Class 6 Maths Chapter 5, question 48 — Source question

1. 2024 is a leap year (as February has 29 days). Leap years occurs in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400. a. From the year you were born till now, which years were leap years? b. From the year 2024 till 2099, how many leap years are there?

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b. From the year 2024 till 2099, the number of leap years is 19

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NCERT Class 6 Maths Chapter 5, question 49 — Source question

2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

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The largest 4-digit number divisible by 4 and also a palindrome: 9999 The smallest 4-digit number divisible by 4 and also a palindrome: 1001.

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NCERT Class 6 Maths Chapter 5, question 50 — Source question

3. Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning. a. Sum of two even numbers gives a multiple of 4. b. Sum of two odd numbers gives a multiple of 4.

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10 a. Sometimes true Example 2+6 = 8, a multiple of 4. but 2+4 = 6, not a multiple of 4. b. Sometimes true Example 1+3 = 4, a multiple of 4. but 1+5 = 6, not a multiple of 4. Note: Think of more such examples.

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Note: Think of more such examples.

NCERT Class 6 Maths Chapter 5, question 51 — Source question

4. Find the remainders obtained when each of the following numbers are divided by i) 10, ii) 5, iii) 2. 78, 99, 173, 572, 980, 1111, 2345

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Numbers a) Remainder when c) Remainder when divided by 10 b) Remainder when divided by 5 divided by 2 78 99 173 572 980 1111 2345 8 9 3 2 0 1 5 3 4 3 2 0 1 0 0 1 1 0 0 1 1

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NCERT Class 6 Maths Chapter 5, question 52 — Source question

5. The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of 14560 by only two of these numbers and then declared that it was also divisible by all of them. What could those two numbers be?

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The two numbers that are sufficient to declare the divisibility of 14560 by 2, 4, 5, 8 and 10 are 5 and 8.

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NCERT Class 6 Maths Chapter 5, question 53 — Source question

6. Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.

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The numbers divisible by 2, 4, 5, 8 and 10 are: 5600, 6000 and 77622160.

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NCERT Class 6 Maths Chapter 5, question 54 — Source question

7. Write two numbers whose product is 10000. The two numbers should not have 0 as their unit digit.

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10000 = 16 × 625. 11

Source worked solution for Class 6 Chapter 5, question 54, part 1 — Source worked answer · part 1

Match, attempt, then check

1. At what number is ‘idli-vada’ said for the 10th time?

2. If the game is played for the numbers from 1 till 90, find out: a. How many times would the children say ‘idli’ (including the times they say ‘idli- vada’)?

3. What if the game was played till 900? How would your answers change?

4. Is this figure somehow related to the ‘idli-vada’ game? Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.

What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all.

1. Is there anything common among the shaded numbers?

2. Is there anything common among the circled numbers?

3. Which numbers are both shaded and circled? What are these numbers called?

1. Find all multiples of 40 that lie between 310 and 410.

2. Who am I? a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8. b. I am a number less than 100. Two of my factors are 3 and 5.One of my digits is 1 more than the other.

3. A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.

4. Find the common factors of: a. 20 and 28 b. 35 and 50 c. 4, 8 and 12 d. 5, 15 and 25

5. Find any three numbers that are multiples of 25 but not multiples of 50.

6. Anshu and his friends play the ‘idli-vada’ game with two numbers, which are both smaller than 10. The first time anybody says ‘idlivada’ is after the number 50. What could the two numbers be which are assigned ‘idli’ and ‘vada’?

7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?

8. In the diagram below, Guna has erased all the numbers except the common multiples. Find out what those numbers could be and fill in the missing numbers in the empty regions.

9. Find the smallest number that is a multiple of all the numbers from 1 to 10 except for 7.

10. Find the smallest number that is a multiple of all the number from 1 to 10.

How many prime numbers are there from 21 to 30? How many composite numbers are there from 21 to 30?

1. We see that 2 is a prime and also an even number. Is there any other even prime?

2. Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?

3. Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes?

4. Which of the following numbers are prime: 23, 51, 37, 26?

5. Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

6. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.

7. Find seven consecutive composite numbers between 1 and 100.

8. Twin primes are pairs of primes having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin primes between 1 and 100.

10. Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

11. How many three-digit prime numbers can you make using each of 2, 4 and 5 once?

12. Observe that 3 is a prime number, and 2 × 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.

Which of the following pairs of numbers are co-prime? a. 18 and 35 b. 15 and 37 c. 30 and 415 d. 17 and 69 e. 81 and 18

1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

2. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?

3. Find three prime numbers, all less than 30, whose product is 1955.

4. Find the prime factorisation of these numbers without multiplying first a. 56 × 25 b. 108 × 75 c. 1000 × 81

5. What is the smallest number whose prime factorisation has: a. three different prime numbers? b. four different prime numbers?

1. Are the following pairs of numbers co-prime? Guess first and then use prime

2. Is the first number divisible by the second? Use prime factorisation. a. 225 and 27 b. 96 and 24 c. 343 and 17 d. 999 and 99 3 × 3 × 5 × 5

3. The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? Does one of them divide the other?

4. Guna says, “Any two prime numbers are co-prime”. Is he right?

Is 8536 divisible by 4?

Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe?

Change the last two digits of 8560 so that the resulting number is a multiple of 8.

2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

3. Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning. a. Sum of two even numbers gives a multiple of 4. b. Sum of two odd numbers gives a multiple of 4.

4. Find the remainders obtained when each of the following numbers are divided by i) 10, ii) 5, iii) 2. 78, 99, 173, 572, 980, 1111, 2345

5. The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of 14560 by only two of these numbers and then declared that it was also divisible by all of them. What could those two numbers be?

6. Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.

7. Write two numbers whose product is 10000. The two numbers should not have 0 as their unit digit.