HCF and LCM

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Description of Levels

Sieve Use the Sieve of Eratosthenes to find prime numbers.

Factor Trees An interactive and very visual way to break down a number into its prime factors.

Level 1 - Finding the highest common factor (HCF) of two numbers.

Level 2 - Finding the lowest common multiple (LCM) of two numbers

Level 3 - Finding the highest common factor (HCF) of large numbers.

Level 4 - Finding the lowest common multiple (LCM) of large numbers

Level 5 - Finding the HCF and LCM of three numbers

Level 6 - Given the HCF and LCM find the numbers

Level 7 - Mixed application questions

HCF and LCM given An Advanced Lesson Starter.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Help

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

You can Find the HCF of numbers by listing the prime factors of both numbers then multiplying together the factors that appear in both lists.

For example find the HCF of 24 and 36
24 = 2x2x2x3 and 36 = 2x2x3x3
so the HCF of 24 and 36 is 2x2x3 = 12

The lowest common multiple (LCM), or least common multiple, is the smallest number that both numbers divide into exactly.

You can Find the LCM of numbers by listing the prime factors of both numbers and then multiply all the prime factors of the larger number by those prime factors of the smaller number that are not already included.

For example find the LCM of 24 and 36
24 = 2x2x2x3 and 36 = 2x2x3x3
so the LCM of 24 and 36 is 2x2x3x3 x 2 = 72

Venn Diagram

A Venn diagram may help you with the task of finding the HCF and LCM of 24 and 36.

Express each number as the primes which multiplied together would give you that number. Write them in Venn diagram sets:

Show the sets intersecting

Multiply the numbers in the intersection of the sets to find the HCF, 2x2x3 = 12.

Multiply all the numbers in the overlapping sets diagram to find the LCM, 2x2x2x3x3 = 72.

The Indian Method

Click here to see an animated demonstration of this cool way to find both the HCF and LCM of two numbers.

A Calculator Method

Advanced calculators have built in functions for finding the HCF and LCM of two numbers but there is a trick for finding the HCF using a modern scientific calculator.

If the two numbers are entered using the fraction template the calculator will express that fraction in its lowest terms. It does this by dividing numerator and denominator by their HCF.

For example to find the HCF of 24 and 36 enter 24/36 then press enter.

Considering the denominators, we now need to find what 24 was divided by to give 2. So dividing 24 by 2 gives 12 which is the HCF.

Connecting HCF and LCM

When you have found the HCF of the numbers a and b the LCM can be found using the following formula:

LCM = ab ÷ HCF

Greatest Common Divisor

It is worth knowing that HCF is also known as GCD. If you are using a spreadsheet such as Excel there are functions named LCM and GCD for calculating the LCM and HCF.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

1. Mr Bytheway was celebrating his birthday. He noticed that his age was divisible by 9 and 21. How old was Mr Bytheway?
2. Len has been baking cakes for the school Fun Day Fete. He has made 144 Iced buns and 240 chocolate chip muffins. He wants to create some identical refreshment tables that will operate during the day. Each table will have the same number of each type of cake. What is the greatest number of refreshment tables that Len can stock?	tables
3. The 17-year locust (Magicicada) can only be seen every 17 years. The 13-year locust only appears every 13 years. If they were both seen in the year 2016, in which year would they next both appear together?
4. Playtime Playground Equipment Ltd. would like to donate 161 swings and 46 roundabouts to parks around the country. The company would like to make sure that each park receives the same number of swings and roundabouts with none left over. What is the greatest number of parks that the company can donate to?	parks
5. Every 33 seconds I hear a ding Every 22 seconds I hear a dong I started a timer when I heard dong-ding To hear it again I must wait how long?	seconds
6. Noreen and Doreen were flower arrangers. Noreen included 54 flowers in each of her arrangements while Doreen included 72 flowers in each of hers. At the end of a month they realised that they had both used the same total number of flowers in the arrangements they had made. What is the smallest that total could have been?	flowers
7. At a District Scout Camp the Scouts could form teams of 7, 14 or 20 with no Scout left out. What was the least number of Scouts that could have been at the camp?	Scouts
8. Three pieces of string have lengths 124cm, 286cm and 320cm. They are each to be cut into pieces of the same length so that no string is left over. What is the longest this length could be?	cm
9. Tess O'late has a rectangular canvas measuring 240cm by 150cm. She wants to fill the canvas with painted tessellating squares. What is the smallest number of squares she must paint to fill the canvas?	squares
10. There are stepping stones across Silver Stream. The stones are numbered from one to fifty three. The green frog starts on stone number one and crosses the stream by jumping to every other stone. The brown frog starts on stone number one but jumps over two stones to land on the next as it crosses the stream. The mighty teal frog starts on stone number one but jumps over six stones to land on the next as it crosses the stream. Which stone (other than stone number one) will all three frogs visit?
11. Nora, Dora and Jora are three friends who do not know much about healthy eating. Nora had £8.15 Dora had £10.07 Jora had £6.95 Nora, Dora and Jora each bought as many gobstoppers as they could with their money. They realised that they each then had exactly 23p left over. What is the greatest possible cost of a gobstopper?	p
12. What is the smallest whole number that is exactly divisible by two thirds, four fifths and one half?

HCF and LCM

Practise finding the highest common factor (HCF), sometimes called the greatest common divisor, and the lowest common multiple (LCM) of two numbers.

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