Class 6 Maths Chapter 5 Prime Time – easy notes, concepts, NCERT book solutions, important questions, worksheet and answers. Learn about prime numbers, composite numbers, factors, multiples, divisibility and prime factorisation with simple explanations.
📘 Class 6 Mathematics – Prime Time
Ganita Prakash | Chapter 5 | Complete Concept Notes & Easy Solutions
🌟 Chapter at a Glance
This chapter helps us understand factors, multiples, common factors, common multiples, prime numbers, composite numbers, co-prime numbers, prime factorisation and divisibility tests.
We also learn how these ideas can be used to solve number puzzles and mathematical games.
📚 Concept Notes
1. Factors
A factor of a number divides that number exactly without leaving a remainder.
Therefore, 4 is a factor of 12.
Factors of 12 are: 1, 2, 3, 4, 6 and 12.
2. Multiples
Multiples are obtained by multiplying a number by 1, 2, 3, 4, and so on.
3. Common Multiples
Numbers that are multiples of two or more numbers are called common multiples.
Multiples of 5 = 5, 10, 15, 20, 25, …
Common multiples = 15, 30, 45, …
4. Common Factors
Factors that are common to two or more numbers are called common factors.
Factors of 18 = 1, 2, 3, 6, 9, 18
Common factors = 1, 2, 3, 6
5. Prime Numbers
A prime number has exactly two factors: 1 and the number itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19…
6. Composite Numbers
A composite number has more than two factors.
Examples: 4, 6, 8, 9, 10, 12…
The number 1 is neither prime nor composite because it has only one factor.
7. Co-prime Numbers
Two numbers are co-prime if they have no common factor other than 1.
8. Prime Factorisation
Writing a number as a product of prime numbers is called prime factorisation.
The prime factors of 56 are 2 and 7.
9. Divisibility Rules
- Divisible by 10: The number ends in 0.
- Divisible by 5: The number ends in 0 or 5.
- Divisible by 2: The last digit is 0, 2, 4, 6 or 8.
- Divisible by 4: The number formed by the last two digits is divisible by 4.
- Divisible by 8: The number formed by the last three digits is divisible by 8.
5.1 Common Multiples and Common Factors
‘Idli-vada’ is said at common multiples of 3 and 5.
a. How many times would the children say ‘idli’ (including the times they say ‘idli-vada’)?
b. How many times would the children say ‘vada’ (including the times they say ‘idli-vada’)?
c. How many times would the children say ‘idli-vada’?
a.
Multiples of 3 up to 90: 90 ÷ 3 = 30
So, children say ‘idli’ 30 times, including ‘idli-vada’.
b.
Multiples of 5 up to 90: 18 multiples.
So, children say ‘vada’ 18 times, including ‘idli-vada’.
c.
Common multiples of 3 and 5 are multiples of 15.
90 ÷ 15 = 6
Therefore, ‘idli-vada’ is said 6 times.
Number of multiples of 3:
Number of multiples of 5:
Number of common multiples:
Therefore:
- ‘idli’ = 300 times
- ‘vada’ = 180 times
- ‘idli-vada’ = 60 times
Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.
Yes. The figure is related to the common multiples of 3 and 5.
The common numbers are the numbers at which we say ‘idli-vada’.
One number is 4. We need a number such that the multiples of 4 and that number do not produce only ‘vada’ numbers separately.
The answer given in the chapter solution is: 8.
We need common factors of 15 and 30.
Factors of 15 = 1, 3, 5, 15.
Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30.
Common factors are: 1, 3, 5 and 15.
Yes. All the shaded numbers are multiples of 3.
Yes. All the circled numbers are multiples of 4.
The numbers are: 36, 48 and 60.
They are called common multiples of 3 and 4.
Multiples of 40 near this range are:
Therefore: 320, 360, 400.
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.
a.
A number below 40 having factor 7 can be 7, 14, 21, 28 or 35.
Only 35 has digit sum: 3 + 5 = 8.
Answer: 35.
b.
The number must be a multiple of both 3 and 5.
45 is less than 100 and its digits differ by 1: 5 − 4 = 1.
Answer: 45.
Try the numbers between 1 and 10.
Therefore: 6 is a perfect number.
a. 20 and 28
b. 35 and 50
c. 4, 8 and 12
d. 5, 15 and 25
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.
Multiples of 25 include: 25, 50, 75, 100, 125, 150, …
We need numbers that are not multiples of 50.
Examples: 25, 75, 125.
We need two numbers less than 10 whose first common multiple is greater than 50.
Examples given in the chapter solution are:
7 and 8
and
8 and 9.
We need the common factors of 28 and 70.
Factors of 28: 1, 2, 4, 7, 14, 28.
Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70.
Common factors: 1, 2, 7 and 14.
Multiples of ____
Common multiples
72
48
24
Multiples of ____
The common multiples shown are: 24, 48 and 72.
A suitable pair is: Multiples of 3 and Multiples of 4.
More examples can also be explored by checking which two numbers have 24, 48 and 72 as common multiples.
We need the smallest number divisible by: 1, 2, 3, 4, 5, 6, 8, 9 and 10.
The required number is: 360.
The smallest number divisible by every number from 1 to 10 is:
2520.
5.2 Prime Numbers
Numbers from 21 to 30: 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.
Prime numbers: 23 and 29.
Therefore, there are 2 prime numbers.
The remaining 8 numbers are composite.
Therefore, there are 8 composite numbers.
No.
2 is the only even prime number.
Every other even number has 2 as a factor and therefore has more than two factors.
Smallest difference:
Largest difference:
Therefore: Smallest = 1, Largest = 8.
No. The number of primes is not equal in every row.
The decade 91 to 100 has the least number of primes.
The decades 1 to 10 and 11 to 20 have the most.
23 has only factors 1 and 23, so it is prime.
37 has only factors 1 and 37, so it is prime.
51 = 3 × 17, so it is composite.
26 = 2 × 13, so it is composite.
Answer: 23 and 37.
Examples:
Thus, three pairs are: (2,3), (3,7), (2,13).
Pairs include:
17 and 71
37 and 73
79 and 97
One suitable set is:
90, 91, 92, 93, 94, 95, 96
Each of these numbers is composite.
Twin-prime pairs between 1 and 100 include:
3 & 5, 5 & 7, 11 & 13, 17 & 19, 29 & 31, 41 & 43, 59 & 61, 71 & 73
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.
a. True. A number ending in 4 is even. The only even prime number is 2. Therefore, a prime number cannot have units digit 4.
b. False. A product of two or more primes is composite.
c. False. A prime number has exactly two factors: 1 and itself.
d. False. 2 is even and prime. Other even numbers are composite.
e. True. After the pair 2 and 3, the next number after a prime is even and greater than 2, so it is composite.
Check 105:
3, 5 and 7 are three different prime numbers.
Therefore: 105.
Every arrangement uses 2, 4 and 5.
Every such number is even or ends in 5. Therefore, every arrangement is divisible by 2 or 5.
So none of them can be prime.
Answer: None.
Examples:
Thus, examples include: 5, 11, 23, 29 and 41.
5.3 Co-prime Numbers
a. 15 and 39
b. 4 and 15
c. 18 and 29
d. 20 and 55
A safe pair has no common factor other than 1.
15 and 39 have common factor 3, so they are not safe.
4 and 15 have no common factor other than 1, so they are safe.
18 and 29 have no common factor other than 1, so they are safe.
20 and 55 have common factor 5, so they are not safe.
Answer: 4 and 15; 18 and 29.
a. 18 and 35
b. 15 and 37
c. 30 and 415
d. 17 and 69
e. 81 and 18
The co-prime pairs are:
18 and 35
15 and 37
17 and 69
The other pairs have a common factor.
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?
1. When the first common multiple equals the product:
Examples: 3 and 5; 3 and 7; 4 and 9.
2. When the first common multiple is less than the product:
Examples: 3 and 6; 3 and 12; 6 and 15.
The relationship is: when two numbers are co-prime, their first common multiple is the product of the two numbers.
5.4 Prime Factorisation
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
= 198
Therefore:
1955 = 5 × 17 × 23
Answer: 5, 17 and 23.
a. 56 × 25
b. 108 × 75
c. 1000 × 81
a.
56 = 2 × 2 × 2 × 7
25 = 5 × 5
Therefore:
56 × 25 = 2 × 2 × 2 × 7 × 5 × 5
b.
108 = 2 × 2 × 3 × 3 × 3
75 = 3 × 5 × 5
Therefore:
108 × 75 = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5
c.
1000 = 2 × 2 × 2 × 5 × 5 × 5
81 = 3 × 3 × 3 × 3
Therefore:
1000 × 81 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 × 5
a. three different prime numbers?
b. four different prime numbers?
a.
The three smallest different primes are 2, 3 and 5.
2 × 3 × 5 = 30.
b.
The four smallest different primes are 2, 3, 5 and 7.
2 × 3 × 5 × 7 = 210.
a. 30 and 45
b. 57 and 85
c. 121 and 1331
d. 343 and 216
a. No.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
They have common factors 3 and 5.
b. Yes.
57 = 3 × 19
85 = 5 × 17
There is no common prime factor.
c. No.
121 = 11 × 11
1331 = 11 × 11 × 11
They have common factor 11.
d. Yes.
343 = 7 × 7 × 7
216 = 2 × 2 × 2 × 3 × 3 × 3
There is no common prime factor.
a. 225 and 27
b. 96 and 24
c. 343 and 17
d. 999 and 99
a. No.
225 = 3 × 3 × 5 × 5
27 = 3 × 3 × 3
225 does not contain three 3s.
b. Yes.
96 = 2 × 2 × 2 × 2 × 2 × 3
24 = 2 × 2 × 2 × 3
The prime factors of 24 are included in those of 96.
c. No.
343 = 7 × 7 × 7
17 is not a factor.
d. No.
999 = 3 × 3 × 3 × 37
99 = 3 × 3 × 11
999 does not contain factor 11.
Both numbers contain common prime factors 3 and 7.
Therefore, they are not co-prime.
The first number does not contain factor 11, so the second number does not divide the first.
The first number also contains factor 2, so the first does not divide the second.
Therefore, neither number divides the other.
Yes.
Two different prime numbers have only 1 as their common factor.
For example:
2 and 3 are co-prime.
3 and 11 are co-prime.
5.5 Divisibility Tests
Look at the last two digits:
36 is divisible by 4 because:
Therefore: 8536 is divisible by 4.
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?
Yes, all three statements are true.
For divisibility by 4, we only need to check the last two digits.
Therefore numbers such as 124, 364 and 4028 are divisible by 4.
Between 120 and 140:
128, 136
Between 1120 and 1140:
1128, 1136
Between 3120 and 3140:
3128, 3136
Observation: the last three digits determine divisibility by 8.
One suitable number is:
Therefore: 8552 is a multiple of 8.
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?
Yes, all three statements are true.
For divisibility by 8, we check the last three digits.
Examples include: 8576, 7648 and 5024.
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?
For part (b), the years divisible by 4 from 2024 to 2099 are counted. There are:
19 leap years.
Part (a) depends on the year of birth of the student.
A palindrome reads the same from left to right and right to left.
The chapter solution gives:
Largest: 9999
Smallest: 1001
a. Sum of two even numbers gives a multiple of 4.
b. Sum of two odd numbers gives a multiple of 4.
a. Sometimes true.
b. Sometimes true.
78, 99, 173, 572, 980, 1111, 2345
| Number | ÷ 10 | ÷ 5 | ÷ 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 |
The two useful tests are divisibility by 5 and 8.
14560 ends in 0, so it is divisible by 5.
The last three digits are 560.
So it is divisible by 8.
Since 8 implies divisibility by 2 and 4, and divisibility by 5 together with an even number gives divisibility by 10, the required two numbers are:
5 and 8.
Check the numbers using the divisibility rules.
The numbers divisible by all five are:
5600, 6000 and 77622160
Neither 16 nor 625 has 0 as its units digit.
Answer: 16 and 625.
5.6 Fun with Numbers
⭐ Special Numbers
The chapter gives the numbers:
Different numbers can be considered special for different reasons. For example, 43 is prime and is also the only number in the box that is not a square.
🧩 Prime Puzzle
The rule of the prime puzzle is:
The displayed solution in the textbook uses prime numbers such as 2, 3, 5, 7 and 17 to obtain the required row and column products.
📝 Quick Revision
- A factor divides a number exactly.
- Multiples are obtained by multiplication.
- Common multiples belong to both sets of multiples.
- Common factors divide both numbers exactly.
- A prime number has exactly two factors.
- 1 is neither prime nor composite.
- A composite number has more than two factors.
- Co-prime numbers have no common factor other than 1.
- Prime factorisation expresses a number as a product of primes.
- For divisibility by 10, check the last digit.
- For divisibility by 5, check whether the last digit is 0 or 5.
- For divisibility by 2, check whether the last digit is even.
- For divisibility by 4, check the last two digits.
- For divisibility by 8, check the last three digits.
Practice Class 6 Maths Chapter 5 with this easy and engaging worksheet. This worksheet is useful for practice, revision and exam preparation.
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