Factors and Multiples
When two or more numbers are multiplied to get the product, then each number is a factor of the product and the product is a multiple of each of those numbers.
- Prime Numbers
- Composite Numbers
- Prime Numbers between 1 and 100
- Prime Factorisation
- Common Factors
- Highest Common Factor (HCF)
- Common Multiples
- Lowest Common Multiples (LCM)
- Find HCF by Prime Factorisation
- Find LCM by Prime Factorisation
Prime Numbers
| Product | Multiplication facts | Factors |
|---|---|---|
| 2 | 1 x 2 = 2 | 1, 2 |
| 3 | 1 x 3 = 3 | 1, 3 |
| 4 | 1 x 4 = 4 2 x 2 = 4 | 1, 2, 4 |
| 5 | 1 x 5 = 5 | 1, 5 |
| 6 | 1 x 6 = 6 2 x 3 = 6 | 1, 2, 3, 6 |
| 7 | 1 x 7 = 7 | 1, 7 |
| 8 | 1 x 8 = 8 2 x 4 = 8 | 1, 2, 4, 8 |
| 9 | 1 x 9 = 9 3 x 3 = 9 | 1, 3, 9 |
| 10 | 1 x 10 = 10 2 x 5 = 10 | 1, 2, 5, 10 |
We can see that the numbers 2, 3, 5 and 7 have only two factors (1 and the number itself). This type of numbers are called
Composite Numbers
From the above table, we can see that the numbers 4, 6, 8 and 10 have more than 2 factors. They are called composite numbers.
1 has only one factor (1 itself). So, 1 is neither a prime nor a composite.
Prime Numbers between 1 and 100
The below table will help you in identifying the prime and composite numbers between 1 and 100.
All the prime numbers are highlighted in
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
The prime numbers between 1 and 100 are
» There are 74 composite numbers between 1 and 100.
» 1 is neither a prime nor a composite.
» 2 is the one and only one number, that is both prime and even.
Watch the below video to learn how you can prepare the above table.
Prime Factorisation
Expressing a number as the product of its factors is called
Expressing a number as the product of its prime factors is called
24 = 2 x 12
= 2 x 2 x 6
= 2 x 2 x 2 x 3
We have expressed
| Number | Prime Factorisation |
|---|---|
| 30 | 2 x 3 x 5 |
| 48 | 2 x 2 x 2 x 2 x 3 |
| 100 | 2 x 2 x 5 x 5 |
To convert a number into the product of its prime factors, we divide the given number by the prime numbers 2, 3, 5, 7, 11 etc, in the given order repeatedly.
Let's divide 420 with possible prime numbers from small to big.
2 210
3 105
5 35
7
7 is a prime and we have arrived with the last possible prime factor. So, the prime factorisation of
420 = 2 x 2 x 3 x 5 x 7
Common Factors
We know, the factors of
Factors of
The
We will learn the importance of common factors in the subsequent sessions.
Highest Common Factor (HCF)
From the above example, we have seen that the
Among these factors, 6 is the gratest. 6 is known as the
| Numbers | Factors | Common Factors | HCF |
|---|---|---|---|
| 24, 40 | 24: 1, 2, 3, 4, 6, 8, 12, 24 40: 1, 2, 4, 5, 8, 10, 20, 40 | 1, 2, 4, 8 | 8 |
| 20, 64 | 20: 1, 2, 4, 5, 10, 20 64: 1, 2, 4, 8, 16, 32, 64 | 1, 2, 4 | 4 |
| 25, 45 | 25: 1, 5, 25 45: 1, 3, 5, 9, 15, 45 | 1, 5 | 5 |
Common Multiples
Let's consider the numbers
The multiples of
The multiples of
We can see that some multiples are common. They are
Lowest Common Multiples (LCM)
From the above example, we have seen that
| Numbers | Multiples | Common Multiples | LCM |
|---|---|---|---|
| 4, 10 | 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... 10: 10, 20, 30, ... | 20, 40, ... | 20 |
| 5, 6, 15 | 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ... 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ... 15: 15, 30, 45, 60, ... | 30, 60, ... | 30 |
| 4, 8 | 4: 4, 8, 12, 16, ... 8: 8, 16, ... | 8, 16 | 8 |
» When one number is the multiple of the other, then HCF of those numbers will be always the smallest number.
» The LCM of a group of numbers cannot be less than the greatest number in that group.
Find HCF by Prime Factorisation
To find the HCF of a group of numbers, we express each one of them as the product of its prime factors. Then we multiply all the
2 28, 72
2 14, 36
7, 36
We have arrived in a step with numbers 7 and 36, in which we cannot have a common factor.
HCF = product of common prime factors.
Hence, HCF of 28 and 72 = 2 x 2 = 4
7 14, 84, 105
2, 12, 15
We have arrived in a step with numbers 2, 12 and 15, in which we cannot have a common factor.
HCF = product of common prime factors.
Hence, HCF of 14, 84 and 105 = 7
2 12, 48, 96
2 6, 24, 48
3 3, 12, 24
1, 4, 8
We have arrived in a step with numbers 1, 4 and 8, in which we cannot have a common factor.
HCF = product of common prime factors.
Hence, HCF of 12, 48 and 96 = 2 x 2 x 3 = 12
Find LCM by Prime Factorisation
To find the LCM of a group of numbers, we express each one of them as the product of its prime factors. Then we multiply
5 15, 40
3, 8
We have arrived in a step with numbers 3 and 8, in which we cannot have a common factor.
LCM = product of all the prime factors.
Hence, LCM of 15 and 40 = 5 x 3 x 8 = 120
2 16, 48, 192
2 8, 24, 96
2 4, 12, 48
2 2, 6, 24
3 1, 3, 12
1, 1, 4
We need to continue the process of finding the prime factors, if there is a common prime factor for at least two numbers.
We have arrived in a step with numbers 1, 1 and 4, in which we cannot have a common factor, other than 1.
LCM = product of all the prime factors.
Hence, LCM of 16, 48 and 192
= 2 x 2 x 2 x 2 x 3 x 1 x 1 x 4 = 192