**Factors,
multiples and prime factorisation**

**Level 7**

The objectives of this unit are to

* draw factor trees in order to obtain the prime factorisation of a number;

* to find the lowest common multiple and the highest common factor of two numbers.

* * * * *

**Basic concepts**

The **factors** of a whole number are the numbers that divide into it
exactly.

For example, the factors of 18 are 1, 2, 3, 6, 9, and 18.

A number is called a **prime number** if it has exactly two
factors (i.e. 1 and itself). The first
few prime numbers are 2, 3, 5, 7, 11, 13, … .
Note that 1 is not a prime number.

The **multiples** of a number are all the numbers that it will go into
exactly. For example, the multiples of 6
are 6, 12, 18, 24, 30, … .

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**Factor trees**

Factor trees are used to break a number up into its prime factors.

**Example**: Draw the factor
tree for 60.

We begin by finding two whole numbers that multiply to make 60, for example 6 × 10:

We keep splitting up each number in this way until we reach a prime number. Once a prime number is reached, we circle it and then that part of the diagram is finished. The completed factor tree looks as follows:

The circled numbers multiply together to make 60, i.e. 60 = 2 × 2 × 3 × 5.

This product is called the **prime factorisation** of 60.

It is best to write the prime
factorisation using powers: 60 = 2^{2}
× 3 × 5.

**Example 2**: Find the prime
factorisation of 72.

**Solution**: The factor tree
for 72 is as follows:

So, the prime factorisation of 72
is: 72 = 2 × 2 × 2 × 3 × 3 OR 2^{3}
× 3^{2}.

* * * * *

**Highest common factor**

The highest common factor (HCF) of 2 numbers is the largest number that divides exactly into both numbers.

One way to find the HCF is to list all the factors of both numbers.

**Example**: Find the highest
common factor of 45 and 63.

**Solution**: The factors
of 45 are 1, 3, 5, 9, 15 and 45

The factors of 63 are 1, 3, 7, 9, 21 and 63.

The common factors therefore are 1, 3 and 9. So the highest common factor is 9.

* * * * *

**Lowest common multiple**

The lowest common multiple (LCM) of 2 numbers is the smallest number that is a multiple of both of them.

One way to find the LCM is to list out the multiples of both numbers.

**Example**: Find the lowest
common multiple of 6 and 8.

**Solution**: The
multiples of 6 are 6, 12, 18, 24, 30,
36, 42, 48, 54, 60,…

The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

The common multiples are 24, 48, 72, … So the lowest common multiple is 24.

* * * * *

**Find the HCF and LCM using prime factors**

The above methods for finding a HCF or a LCM become increasingly more difficult as the numbers become larger. Another method of finding the HCF and the LCM is to use factor trees.

**Example**: Find the highest
common factor and lowest common multiple of 84 and 140.

**Solution**:

__Step 1__: Draw factor trees for both numbers and write
out the prime factorisations.

The factor tree for 84 is:

The factor tree for 140 is:

So:
84 = 2 × 2 × 3 × 7 = 2^{2} × 3 × 7

Also, 140 = 2 × 2 × 5 × 7 = 2^{2} × 5 × 7

__Step 2__: Put the circled numbers (the prime factors
into a Venn diagram):

__Step 3__: To find the highest common factor of 84 and
140, you multiply together the numbers in the overlap:

HCF = 2 × 2 × 7 = 28

__Step 4__: To find the LCM, you multiply together all
the numbers INSIDE the Venn diagram:

LCM = 3 × 2 × 2 × 7 × 5 = 420.