# Unit 2 Section 4 : Highest Common Factor and Lowest Common Multiple

This section introduces the idea of the highest common factor and lowest common multiple of a pair of numbers.

 The highest common factor (HCF) of two whole numbers is the largest whole number which is a factor of both.

## HCF Example

Consider the numbers 12 and 15:
The factors of 12 are : 1, 2, 3, 4, 6, 12.
The factors of 15 are : 1, 3, 5, 15.
1 and 3 are the only common factors (numbers which are factors of both 12 and 15).

Therefore, the highest common factor of 12 and 15 is 3.

Example Questions
Work out the answers to these questions and click the buttons marked to see whether you are correct.

(a) Find the highest common factor of 20 and 30 by following the steps below:
 What are the factors of 20? What are the factors of 30? What is the highest common factor of 20 and 30?

(b) Find the highest common factor of 14 and 12 by following the steps below:
 What are the factors of 14? What are the factors of 12? What is the highest common factor of 14 and 12?

 The lowest common multiple (LCM) of two whole numbers is the smallest whole number which is a multiple of both.

## LCM Example

Consider the numbers 12 and 15 again:
The multiples of 12 are : 12, 24, 36, 48, 60, 72, 84, ....
The multiples of 15 are : 15, 30, 45, 60, 75, 90, ....
60 is a common multiple (a multiple of both 12 and 15), and there are no lower common multiples.

Therefore, the lowest common multiple of 12 and 15 is 60.

Example Questions
Work out the answers to these questions and click the buttons marked to see whether you are correct.

(a) Find the lowest common multiple of 5 and 7 by following the steps below:
 What are the first ten multiples of 5? What are the first ten multiples of 7? What is the lowest common multiple of 5 and 7?

(b) Find the lowest common multiple of 6 and 10 by following the steps below:
 What are the first ten multiples of 6? What are the first ten multiples of 10? What is the lowest common multiple of 6 and 10?

 Although the methods above work well for small numbers, they are more difficult to follow with bigger numbers. Another way to find the highest common factor and lowest common multiple of a pair of numbers is to use the prime factorisations of the two numbers.

## Finding HCF & LCM with prime factorisations

We want to find the HCF and LCM of the numbers 60 and 72.

Start by writing each number as a product of its prime factors.
 60 = 2 * 2 * 3 * 5 72 = 2 * 2 * 2 * 3 * 3

To make the next stage easier, we need to write these so that each new prime factor begins in the same place:
 60 = 2 * 2 * 3 * 5 72 = 2 * 2 * 2 * 3 * 3
All the "2"s are now above each other, as are the "3"s etc. This allows us to match up the prime factors.

The highest common factor is found by multiplying all the factors which appear in both lists:

So the HCF of 60 and 72 is 2 × 2 × 3 which is 12.

The lowest common multiple is found by multiplying all the factors which appear in either list:

So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.

Example Questions

(a) Find the highest common factor and lowest common multiple of 50 and 70 by following the steps below:
 What is 50 as a product of its prime factors? What is 70 as a product of its prime factors? What is the HCF of 50 and 70? (find it by multiplying together the factors which appear in both lists above) What is the LCM of 50 and 70? (find it by multiplying together the factors which appear in either list above)

(b) Find the highest common factor and lowest common multiple of 900 and 270 by following the steps below:
 What is 900 as a product of its prime factors? What is 270 as a product of its prime factors? What is the HCF of 900 and 270? (find it by multiplying together the factors which appear in both lists above) What is the LCM of 900 and 270? (find it by multiplying together the factors which appear in either list above)

## Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.

Question 1
Find the HCF of 21 and 35 by following the steps below:
(a) List the factors of 21 in the box below. Separate the factors with commas, like this:

(b) List the factors of 35 in the box below.

(c) What is the HCF of 21 and 35?

Question 2
Find the HCF of:
 (a) 6 and 9 : (b) 14 and 18 : (c) 30 and 24 : (d) 15 and 10 :
Question 3
Find the HCF of 42 and 90 by following the steps below:
(a) Write 42 as a product of its prime factors (you might find it helpful to use a factor tree).

(b) Write 90 as a product of its prime factors in the box below:

(c) What is the HCF of 42 and 90?

Question 4
Find the HCF of:
 (a) 90 and 120 : (b) 96 and 72 : (c) 56 and 60 : (d) 77 and 50 : (e) 300 and 550 : (f) 320 and 128 :
Question 5
Find the LCM of 8 and 6 by following the steps below:
(a) List the first five multiples of 8 in the box below.

(b) List the first five multiples of 6 in the box below.

(c) What is the LCM of 8 and 6?

Question 6
Find the LCM of:
 (a) 5 and 3 : (b) 9 and 6 : (c) 8 and 10 : (d) 12 and 9 : (e) 15 and 20 : (f) 6 and 11 :
Question 7
Find the LCM of 66 and 40 by following the steps below:
(a) Write 66 as a product of its prime factors (you might find it helpful to use a factor tree).

(b) Write 40 as a product of its prime factors in the box below:

(c) What is the LCM of 66 and 40?

Question 8
Find the LCM of:
 (a) 28 and 30 : (b) 16 and 24 : (c) 20 and 25 : (d) 60 and 50 : (e) 12 and 18 : (f) 21 and 35 :
Question 9
Two lighthouses can be seen from the top of a hill.
The first flashes once every 8 seconds, and the other flashes once every 15 seconds.

If they flash at the same time, how long will it be until they flash at the same time again?
seconds

You have now completed Unit 2 Section 4
 Your overall score for this section is Correct Answers You answered questions correctly out of the questions in this section. Incorrect Answers There were questions where you used the Tell Me button. There were questions with wrong answers. There were questions you didn't attempt.
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