**Factors**

What are the **factors** of a number?

How many factors do the following numbers have? What are they?

(a) 6 (b) 21 (c) 49 (d) 64 (e) 99

**Prime Numbers**

A prime number is a **very special** kind of number. It has exactly 2 factors.

So 7 is a prime number, because its factors are 1 and 7.

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

**Prime Factors**

Any number can be written as a product of its prime factors and there is only one way to do this

e.g. 30 = 2 x 3 x 5

*** There is no other way to write 30 as a product of its prime factors (other than changing the order in which we write them) ***

It is very useful to us to write a number as a product of prime factors so we must learn an effective method that lets us do this.

**Multiples**

The multiple of a number are all the number that we can get by multiplying it by another whole number

So, for example, the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, …

**Lowest Common Multiple **and **Highest Common Factor**

Two very special numbers that mathematicians are often interested in are the **lowest common multiple** of two numbers and the **highest common factor** of two numbers.

The lowest common multiple is the smallest number that is a multiple of both the numbers

The highest common factor is the largest number that is a factor of both the numbers

We can use a nice method which involves **Venn diagrams** to find both the LCM (the lowest common multiple) and the HCF (the highest common factor) of a pair of numbers.

**Practice Question Set 1**

- Write down all the factors of the following numbers:
- 10
- 12
- 13
- 15
- 17
- 18
- 25
- 30
- 36
- 48

- Write down the first ten multiples of each of the following numbers:
- 3
- 4
- 6
- 7
- 8
- 9
- 12
- 16

- Write down the lowest common multiple of each of the following number pairs:
- 3, 7
- 4, 9
- 6, 8
- 3, 12
- 8, 16

- Write down the highest common factor of each of the following number pairs:
- 12, 18
- 10, 15
- 13, 17
- 36, 48
- 25, 30

- Find the LCM of these pairs of numbers:
- 24, 68
- 180, 420
- 108, 360
- 34, 39
- 180, 450
- 150, 490

- Find the HCF of these pairs of numbers:
- 180, 300
- 270, 378
- 324, 486
- 108, 540
- 450, 990
- 330, 910

- Find the HCF and LCM of:
- 18, 20 and 30
- 9, 12 and 16
- 8, 18 and 50

**Practical Applications**

The HCF and LCM can also be useful to us in solving practical problems. Once we have identified which of them is useful to us, we then calculate it using the methods from above.

**Practice Question Set 2** (Exercise 1J from page 18 of year 8 textbook: Questions 4-6)

- Three strings of different lengths: 240cm, 318cm and 426 cm, are to be cut into equal lengths. What is the greatest possible length of each piece?
- Two lighthouses flash their lights every 20 seconds and 30 seconds respectively. Given that they flashed together at 7pm, when will they next flash together?
- A man has a garden measuring 84m by 56m. He divides it into the minimum number of square plots. What is the length of each square plot?

**Answers**

(1.) 2cm, (2.) 7.01pm. (3.) 14m