# HCF and LCM using Factor Trees

Lesson

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. We can use this to work out the highest common factor between two numbers.

Remember, a prime number is a number that can only be divided by 1 and itself.

#### Examples

##### question 1

What is the highest common factor between $18$18 and $27$27?

From this we can tell the HCF is $9$9.

Let's check out another example.

##### question 2

What is the highest common factor between $16$16 and $24$24?

So the HCF is $8$8.

Remember: Always use the highest prime factor.

When we write numbers as the product of their prime factors, we can use what is common between two numbers to find the highest common factor.

#### Example

##### question 3

Evaluate: What is the highest common factor between $420$420 and $225$225?

Think: What are the common prime factors? They are highlighted below.

420 = 2 x 2 x 3 x 5 x 7

225 = 3 x 3 x 5 x 5

Do: HCF = 3 x 5 = 15

## Lowest Common Multiples (LCMs)

We can also use these prime factors to find the lowest common multiple between numbers.

#### Example

##### question 4

Evaluate: What is the lowest common multiple between $2345$2345 and $765$765?

Think

The lowest common multiple is the smallest product of prime factors that contains all the prime factors of the two numbers. If there are common prime factors between the two numbers, you only have to include them once.

Do: LCM = $5\times7\times67\times3\times3\times17=358785$5×7×67×3×3×17=358785

#### Worked Examples

##### Question 1

Consider the following list of prime factors:

$6125=5\times5\times5\times7\times7$6125=5×5×5×7×7 and

$420175=5\times5\times7\times7\times7\times7\times7$420175=5×5×7×7×7×7×7.

1. Find the highest common factor of $6125$6125 and $420175$420175 by using the list of prime factors.

##### Question 2

By expanding into products of factors, find the highest common factor of $216$216 and $144$144.

1. Write $216$216 as a product of its prime factors.

2. Write $144$144 as a product of its prime factors.

3. Using the product of their prime factors, find the highest common factor of $216$216 and $144$144.

##### Question 3

By expanding into products of factors, find the lowest common multiple of 1750 and 400.

1. Write $1750$1750 as a product of its prime factors.

2. Write $400$400 as a product of its prime factors.

3. Using the product of their prime factors, find the lowest common multiple of $1750$1750 and $400$400.