**Factors**

40 ÷ 8 = 5 and 40 ÷ 5 = 8.

We say that 8 and 5 are factors of 40.

If we can divide a whole number by another whole number without remainder, the second number is a **factor **of the first.

The numbers 1, 2, 4, 5, 8, 10, 20 and 40 all divide into 40. They are all factors of 40. We can write 40 as a product of two factors in eight ways:

40 = 1 x 40 = 2 x 20 = 4 x 10 = 5 x 8

= 8 x 5 = 10 x 4 = 20 x 2 = 40 x 1

**Prime numbers**

A prime number has only two factors, itself and 1.

2, 3, 5, 7, 11, 13, … are prime numbers. 1 has only one factor, itself; 1 is not a prime number.

**Prime factors**

The prime factors of a number are the factors of the number that are prime. It is possible to write every non-prime number as a product of prime factors. For example:

15 = 3 x 5

24 = 2 x 2 x 2 x 3

42 = 2 x 3 x 7

To find the prime factors of a number:

- Start with the lowest prime number, 2.
- Find out if this will divide into the number. If it will not divide, try the next prime number, 3. And so on, trying 5, 7, 11, 13, …. in turn.
- If a prime number will divide, check if it will divide again before moving on to the next prime.

Example

Express 15 288 as a product of prime factors.

15 288 = 2 x 7644

= 2 x 2 x 3822

= 2 x 2 x 2 x 1911

= 2 x 2 x 2 x 3 x 637

= 2 x 2 x 2 x 3 x 7 x 91

= 2 x 2 x 2 x 3 x 7 x 7 x 13

The working can be set out as a continued division as follows:

2 | 15 288

2 | 7 644

2 | 3 822

3 | 1 911

7 | 637

7 | 91

13 | 13

…………….

1

Thus 15 288 = 2 x 2 x 2 x 3 x 7 x 7 x 13

Index form

In the previous example we saw that

15 288 = 2 x 2 x 2 x 3 x 7 x 7 x 13.

It is possible to write the product in a shorter way. 2^{3 }is a short way of writing 2 x 2 x 2. The number 3 in 2^{3 }is called the** index** or **power. ** It shows that three 2’s are to be multiplied together.

In the same way, 7^{2 }is a short way of writing 7 x 7. The index two shows that two 7’s are to be multiplied together.

In index form,

15 288 = 2^{3 }x 3 x 7^{2 }x 13.

The use of index form saves space and can help to prevent errors in counting and copying. It would be easy to make a mistake when copying the products of factors of 256.

256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

In index form, 256 = s^{8}. This saves space and clearly shows that eight 2’s are to be multiplied together.

We say 2^{8 }as ‘two to the power of eight’, or ‘two to the eighth power’ or ‘two raised to the power of eight’ or, often, just ‘two to the eighth’.

The plural of index is indices. The indices 2 and 3are usually said in a special : 7^{2 }as ‘**seven** squared’, 2^{3 }as ‘**two** cubed’.

**Common factors**

The number 12, 21 and 33 are all divisible by 3. We say that 3 is a **common factor** 0f 12, 21 33.

There may be more than one common factor of a set of numbers. For example, both 2 and 7 are common factors of 28, 42 and 70. Since 2 and 7 are common factors and are both prime numbers, then 14 (= 2 x 7) must also be a common factors of the set of numbers.

1 is a common factor of all numbers.

**Highest Common Factor (HCF)**

2, 7 and 14 are common factors of 28, 42 and 70; 14 is the greatest of three common factors. We say that 14 is the **highest common factor** of 28, 42 and 70.

To find the HCF of a set of numbers:

Express the number as a product of prime factors;

- Find the common prime factors
- Multiply the current prime factor together to give the HCF.

**Example**

Find the HCF of 18, 24 and 42.

18 = 2 x 3 x 3

24 = 2 x 2 x 2 3

42 = 2 x 3 x 7

The common prime factors are 2 and 3.

The HCF = 2 x 3 = 6.

Find the HCF of 216 and 288

2 | 216 2 | 288

2 | 108 2 | 144

2 | 54 2 | 72

3 | 27 2 | 36

3 | 9 2 | 18

3 | 3 3 | 9

…… 3 | 3

1 ……..

1

In index notation

216 = 2^{3 }x 3^{3 }

288 = 2^{5 }x 3^{2 }

2^{3 }is the lowest power of two contained in the two numbers. Thus the HCF contains 2^{3}.

3^{2} is the lowest power of 3 contained in the tow numbers. The HCF contains 3^{2}.

216 = (2^{3 }x 3^{3}) x 3

288 = (2^{2 }x 3^{3}) x 2^{2 }

The HCF = 2^{2 }x 3^{3 }= 8 x 6 = 72.

**Multiples**

**Common multiples**

6 is a factor of 12, 18, 24, 30, …We say that 12, 18, 24, 30, …, etc are all multiples of 6. In the same way, 8, 12, 16, 20, …., etc. are all multiples of 4.

Example

Give three common multiples of

- 3 and 11

Solution

33 is a common multiple of 3 and 11.

Likewise, 66 and 99.

Lowest common multiples (LCM)

30, 60, 90 are all common multiples of 6, 10 and 15. 30 is the lowest number that 6, 10 and 15 will divide into. We say that 30 is the lowest common multiple of 6, 10 and 15.

The LCM of 4, 5 and 6 is 60 (not 120).

- Express the number as a product of prime factors;
- Find the lowest product of factors which contains all the prime factors of the numbers.

Example

Find the LCM of 8, 9 and 12.

8 = 2 x 2 x 2

Any multiple of 8 must contain 2 x 2 x 2.

9 = 3 x 3

Any multiple of 9 must contain 3 x 3.

12 = 2 x 2 x 3

Any multiple of 12 must contain 2 x 2 x 3.

The lowest product containing all the three is

2 x 2 x 2 x 3 x 3

The LCM of 8, 9and 12 is 2 x 2 x 2 x 3 x 3 = 72

**EXERCISES**

Lets see how much you’ve learnt, attach the following answers to the comment below

Find the LCM of the following:

- 4 and 6
- 6 and 8
- 9 and 12
- 8, 10 and 15
- 10, 12 and 15