Chapter 5

NCERT Solutions - Class 6 Mathematics

Chapter 5: Prime Time

Page No. 108

Section 5.1 — Multiples and Factors

Q1. At what number is 'idli-vada' said for the 10th time?[cite: 5]

Ans. 150.[cite: 5]

Q2. If the game is played for the numbers from 1 till 90, find out:[cite: 5]
a. How many times would the children say 'idli' (including the times they say 'idli-vada')?[cite: 5]
b. How many times would the children say 'vada' (including the times they say 'idli-vada')?[cite: 5]
c. How many times would the children say 'idli-vada'?[cite: 5]

Ans.
a. 30 times.[cite: 5]
b. 18 times.[cite: 5]
c. 6 times.[cite: 5]

Q3. What if the game was played till 900? How would your answers change?[cite: 5]

Ans. If the game was played till 900, 'idli-vada' would be said 60 times by the children. 'Vada' will be said 180 times and 'idli' will be said 300 times.[cite: 5]

Q4. 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.[cite: 5]

Ans. Yes, the common numbers represent the numbers when to say 'idli-vada'.[cite: 5]

Page No. 109

Section 5.1 — Multiples and Factors (Contd.)

Q. Which of the following could be the other number: 2, 3, 5, 8, 10?[cite: 5]

Ans. The other number will be 8.[cite: 5]

Page No. 110

Figure it Out

Q. What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all.[cite: 5]

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

Q1. Is there anything common among the shaded numbers?[cite: 5]

Ans. All shaded numbers are multiples of 3.[cite: 5]

Q2. Is there anything common among the circled numbers?[cite: 5]

Ans. All circled numbers are multiples of 4.[cite: 5]

Q3. Which numbers are both shaded and circled? What are these numbers called?[cite: 5]

Ans. 36, 48, 60. These numbers are called common multiples of 3 and 4.[cite: 5]

Q1. Find all multiples of 40 that lie between 310 and 410.[cite: 5]

Ans. Multiples of 40 that lie between 310 and 410 are: 320, 360, 400.[cite: 5]

Q2. Who am I?[cite: 5]
a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8.[cite: 5]
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.[cite: 5]

Ans.
a. 35[cite: 5]
b. 45[cite: 5]

Q3. 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.[cite: 5]

Ans. 6.[cite: 5]

Q4. Find the common factors of:[cite: 5]
a. 20 and 28[cite: 5]
b. 35 and 50[cite: 5]
c. 4, 8 and 12[cite: 5]
d. 5, 15 and 25[cite: 5]

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

Q5. Find any three numbers that are multiples of 25 but not multiples of 50.[cite: 5]

Ans. Some such numbers are: 25, 75, 125, 175.[cite: 5]

Q6. Anshu and his friends play the 'idli-vada' game with two numbers, which are both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be which are assigned 'idli' and 'vada'?[cite: 5]

Ans. 7, 8 or 8, 9.[cite: 5]

Q7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?[cite: 5]

Ans. The jump sizes are 1, 2, 7 or 14.[cite: 5]

Q9. Find the smallest number that is a multiple of all the numbers from 1 to 10 except for 7.[cite: 5]

Ans. 360.[cite: 5]

Q10. Find the smallest number that is a multiple of all the numbers from 1 to 10.[cite: 5]

Ans. 2520.[cite: 5]

Page No. 113

Section 5.2 — Prime and Composite Numbers

Q. How many prime numbers are there from 21 to 30? How many composite numbers are there from 21 to 30?[cite: 5]

Ans.
• Prime numbers from 21 to 30: 2 (23, 29)[cite: 5]
• Composite numbers from 21 to 30: 8[cite: 5]

Page No. 114

Figure it Out

Q1. We see that 2 is a prime and also an even number. Is there any other even prime?[cite: 5]

Ans. No.[cite: 5]

Q2. Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?[cite: 5]

Ans. The smallest difference between two successive primes is 1 (3 - 2 = 1); the largest difference is 8 (97 - 89 = 8).[cite: 5]

Q3. 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?[cite: 5]

Ans. No. The least number of primes occur in the decade 91 to 100, and the most number of primes occur in the decades 1 to 10 and 11 to 20.[cite: 5]

Q4. Which of the following numbers are prime: 23, 51, 37, 26?[cite: 5]

Ans. 23 and 37.[cite: 5]

Q5. Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.[cite: 5]

Ans. Pairs of prime numbers less than 20 whose sum is a multiple of 5 are (2, 3), (3, 7), (2, 13).[cite: 5]

Q6. 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.[cite: 5]

Ans. 17 and 71, 37 and 73, 79 and 97.[cite: 5]

Q7. Find seven consecutive composite numbers between 1 and 100.[cite: 5]

Ans. 90, 91, 92, 93, 94, 95, 96.[cite: 5]

Q8. 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.[cite: 5]

Ans. 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and 73.[cite: 5]

Q9. Identify whether each statement is true or false. Explain.[cite: 5]
a. There is no prime number whose units digit is 4.[cite: 5]
b. A product of primes can also be prime.[cite: 5]
c. Prime numbers do not have any factors.[cite: 5]
d. All even numbers are composite numbers.[cite: 5]
e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.[cite: 5]

Ans.
a. True: Unit digit 4 means an even number. 2 is the only even prime number.[cite: 5]
b. False: Product of primes becomes a composite number.[cite: 5]
c. False: Prime numbers have exactly two factors: 1 and the number itself.[cite: 5]
d. False: All even numbers except 2 are composite numbers.[cite: 5]
e. True: Every pair other than (2, 3) contains an even number which is composite.[cite: 5]

Q10. Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?[cite: 5]

Ans. 105 = 3 × 5 × 7 is the only number which is the product of three distinct prime numbers.[cite: 5]

Q11. How many three-digit prime numbers can you make using each of 2, 4 and 5 once?[cite: 5]

Ans. None.[cite: 5]

Q12. 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.[cite: 5]

Ans. Yes. Examples: 11 = 2 × 5 + 1, 23 = 2 × 11 + 1, 47 = 2 × 23 + 1, 59 = 2 × 29 + 1, 83 = 2 × 41 + 1.[cite: 5]

Page No. 115 - 116

Section 5.3 — Co-Prime Numbers

Q. Check if these pairs are safe: a) 15 and 39, b) 4 and 15, c) 18 and 29, d) 20 and 55.[cite: 5]

Ans. Safe pairs are: 4 and 15, 18 and 29.[cite: 5]

Q. Which of the following pairs of numbers are co-prime?[cite: 5]
a. 18 and 35[cite: 5]
b. 15 and 37[cite: 5]
c. 30 and 415[cite: 5]
d. 17 and 69[cite: 5]
e. 81 and 18[cite: 5]

Ans. The pairs of co-prime numbers are: 18 and 35, 15 and 37.[cite: 5]

Q. Find examples where the first common multiple is equal to vs. less than the product of the two numbers. How is it related to co-primes?[cite: 5]

Ans.
1. Equal to product: (3, 5), (3, 7), (4, 9).[cite: 5]
2. Less than product: (3, 6), (3, 12), (6, 15).[cite: 5]
Whenever a number pair is co-prime, the first common multiple is equal to the product of the two numbers.[cite: 5]

Page No. 120

Section 5.4 — Prime Factorisation

Q1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.[cite: 5]

Ans.
• 64 = 2 × 2 × 2 × 2 × 2 × 2[cite: 5]
• 104 = 2 × 2 × 2 × 13[cite: 5]
• 105 = 3 × 5 × 7[cite: 5]
• 243 = 3 × 3 × 3 × 3 × 3[cite: 5]
• 320 = 2 × 2 × 2 × 2 × 2 × 2 × 5[cite: 5]
• 141 = 3 × 47[cite: 5]
• 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3[cite: 5]
• 729 = 3 × 3 × 3 × 3 × 3 × 3[cite: 5]
• 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2[cite: 5]
• 1331 = 11 × 11 × 11[cite: 5]
• 1000 = 2 × 2 × 2 × 5 × 5 × 5[cite: 5]

Q2. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?[cite: 5]

Ans. 198 = 2 × 3 × 3 × 11.[cite: 5]

Q3. Find three prime numbers, all less than 30, whose product is 1955.[cite: 5]

Ans. 1955 = 5 × 17 × 23.[cite: 5]

Q4. Find the prime factorisation of these numbers without multiplying first:[cite: 5]
a. 56 × 25[cite: 5]
b. 108 × 75[cite: 5]
c. 1000 × 81[cite: 5]

Ans.
a. 56 × 25 = 2 × 2 × 2 × 7 × 5 × 5[cite: 5]
b. 108 × 75 = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5[cite: 5]
c. 1000 × 81 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 × 5[cite: 5]

Q5. What is the smallest number whose prime factorisation has:[cite: 5]
a. three different prime numbers?[cite: 5]
b. four different prime numbers?[cite: 5]

Ans.
a. 2 × 3 × 5 = 30[cite: 5]
b. 2 × 3 × 5 × 7 = 210[cite: 5]

Page No. 122

Figure it Out

Q1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer.[cite: 5]
a. 30 and 45[cite: 5]
b. 57 and 85[cite: 5]
c. 121 and 1331[cite: 5]
d. 343 and 216[cite: 5]

Ans.
a. No (30 = 2 × 3 × 5, 45 = 3 × 3 × 5)[cite: 5]
b. Yes (57 = 3 × 19, 85 = 5 × 17)[cite: 5]
c. No (121 = 11 × 11, 1331 = 11 × 11 × 11)[cite: 5]
d. Yes (343 = 7 × 7 × 7, 216 = 2 × 2 × 2 × 3 × 3 × 3)[cite: 5]

Q2. Is the first number divisible by the second? Use prime factorisation.[cite: 5]
a. 225 and 27[cite: 5]
b. 96 and 24[cite: 5]
c. 343 and 17[cite: 5]
d. 999 and 99[cite: 5]

Ans.
a. No[cite: 5]
b. Yes[cite: 5]
c. No[cite: 5]
d. No[cite: 5]

Q3. 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?[cite: 5]

Ans. No, they are not co-prime. No, neither divides the other.[cite: 5]

Q4. Guna says, "Any two prime numbers are co-prime". Is he right?[cite: 5]

Ans. Yes. For example, 2 and 3; 3 and 11.[cite: 5]

Page No. 124 - 125

Section 5.5 — Divisibility Rules

Q. Is 8536 divisible by 4?[cite: 5]

Ans. Yes, as 36 is divisible by 4.[cite: 5]

Q. 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?[cite: 5]

Ans.
• Between 120 & 140: 128, 136[cite: 5]
• Between 1120 & 1140: 1128, 1136[cite: 5]
• Between 3120 & 3140: 3128, 3136[cite: 5]
Observation: If the number formed by the ones, tens, and hundreds digits is divisible by 8, then the whole number is divisible by 8.[cite: 5]

Q. Change the last two digits of 8560 so that the resulting number is a multiple of 8.[cite: 5]

Ans. 8552 is a multiple of 8.[cite: 5]

Q1. 2024 is a leap year. Leap years occur in years that are multiples of 4, except years divisible by 100 but not 400. From 2024 till 2099, how many leap years are there?[cite: 5]

Ans. There are 19 leap years.[cite: 5]

Q2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.[cite: 5]

Ans.
• Largest 4-digit palindrome divisible by 4: 9999[cite: 5]
• Smallest 4-digit palindrome divisible by 4: 1001[cite: 5]

Q3. Explore if each statement is always true, sometimes true or never true:[cite: 5]
a. Sum of two even numbers gives a multiple of 4.[cite: 5]
b. Sum of two odd numbers gives a multiple of 4.[cite: 5]

Ans.
a. Sometimes true: e.g., 2 + 6 = 8 (multiple of 4), but 2 + 4 = 6 (not a multiple of 4).[cite: 5]
b. Sometimes true: e.g., 1 + 3 = 4 (multiple of 4), but 1 + 5 = 6 (not a multiple of 4).[cite: 5]

Q4. Find remainders when divided by i) 10, ii) 5, iii) 2 for the given numbers:[cite: 5]

[cite: 5] [cite: 5] [cite: 5] [cite: 5] [cite: 5] [cite: 5] [cite: 5]

Number	Remainder by 10	Remainder by 5	Remainder by 2
78	8	3	0
99	9	4	1
173	3	3	1
572	2	2	0
980	0	0	0
1111	1	1	1
2345	5	0	1

Q5. What two numbers are sufficient to declare divisibility of 14560 by 2, 4, 5, 8, and 10?[cite: 5]

Ans. 5 and 8.[cite: 5]

Q6. Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160?[cite: 5]

Ans. 5600, 6000, and 77622160.[cite: 5]

Q7. Write two numbers whose product is 10000. The two numbers should not have 0 as their unit digit.[cite: 5]

Ans. 10000 = 16 × 625.[cite: 5]