HCF, LCM and Prime Factorisation

Highest common factor (HCF)

Some numbers share the same factor. For example $2$2 is a factor of $10$10 and $24$24. We call these common factors.

When we are asked to find the highest common factor (HCF) between two or more numbers, we are being asked what is the biggest number that they can both be divided by and that leaves no remainder (or left overs).

Examples

question 1

Evaluate: What is the highest common factor between $12$12 and $24$24?

Think:  The factors of $12$12 are $1,2,3,4,6$1,2,3,4,6 and $12$12.

The factors of $24$24 are $1,2,3,4,6,8,12$1,2,3,4,6,8,12 and $24$24.

So, $1,2,3,4,6$1,2,3,4,6 and $12$12 are all common factors but $12$12 is the highest common factor because it is the biggest number.

Do: The HCF is $12$12.

Let's look at another example.

question 2

Evaluate: What is the highest common factor between $9$9 and $15$15?

Think: The factors of $9$9 are $1$1, $3$3 and $9$9. The factors of $15$15 are $1,3,5$1,3,5 and $15$15.

Do: The HCF is $3$3.

 

Lowest Common Multiple (LCM)

A multiple is the result of multiplying a number by an integer (for more info see Looking at Multiplication Sets).

Remember, some numbers share multiples. For example, $12$12 is a common multiple of $2$2 and $3$3 because $2\times6=12$2×6=12 and $3\times4=12$3×4=12.

We we are asked to find the lowest common multiple, we are being asked to find the smallest multiple that is shared by two numbers.

Examples

question 1

Evaluate: What is the lowest common multiple of $6$6 and $8$8?

Think: Multiples of $6$6 are: $6,12,18,24$6,12,18,24 and $30$30. Multiples of $8$8 are $8,16,24,32$8,16,24,32 and $40$40.

Since $24$24 is the first common number to appear between the two sets, $24$24 is the lowest common multiple.

Do: LCM = $24$24.

Ok, let's try another one.

question 2

Evaluate: What is the lowest common multiple of $10$10 and $12$12?

Think: Multiples of $10$10 are $10,20,30,40,50$10,20,30,40,50 and $60$60. Multiples of $12$12 are $12,24,36,48$12,24,36,48 and $60$60.

Do: LCM = $60$60.

 

Factor Trees

A factor tree is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime. Once we completely break down a number into its prime factors, we can use them to write it in index notation.

Index notation is a short way of writing a number being multiplied by itself several times. For example, $5\times5\times5$5×5×5 can be written as $5^3$53.

Examples

How can we write $8$8 as a product of its prime factors using index notation?

So we can write $8$8 as  $2\times2\times2$2×2×2. When we write this in index notation, it would be $2^3$23.

 

Let's look at another example. How can we write $81$81 as a product of its prime factors using index notation?

So we can break $81$81 down into $4$4 sets of multiplication by $3$3, so in index notation, we would write this as $3^4$34.

 

Sometimes there is more than one prime factor. Let's look at $36$36 as an example:

$36$36 can be written using index notation as $2^2\times3^2$22×32 .

 

Practice questions

Question 1

Question 2

Question 3