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.

 

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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 = 2² × 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  2³ × 3².

 

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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.

 

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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.

 

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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 = 2² × 3 × 7

 

Also,  140 = 2 × 2 × 5 × 7 = 2² × 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.