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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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This is level 7: Mixed application questions. You can earn a trophy if you get at least 9 correct and you do this activity online.

1. Mr Bytheway was celebrating his birthday. He noticed that his age was divisible by 12 and 14. How old was Mr Bytheway?

Correct Wrong

2. Len has been baking cakes for the school Fun Day Fete. He has made 126 Iced buns and 210 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 Correct Wrong

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 2010, in which year would they next both appear together?


Correct Wrong

4. Playtime Playground Equipment Ltd. would like to donate 231 swings and 66 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 Correct Wrong

Every 75 seconds I hear a ding
Every 50 seconds I hear a dong
I started a timer when I heard dong-ding
To hear it again I must wait how long?

seconds Correct Wrong

6. Noreen and Doreen were flower arrangers. Noreen included 36 flowers in each of her arrangements while Doreen included 48 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 Correct Wrong

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 Correct Wrong

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 Correct Wrong

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?Cancvas

squares Correct Wrong

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?

Correct Wrong

11. Nora, Dora and Jora are three friends who do not know much about healthy eating.

Nora had £8.34
Dora had £10.34
Jora had £7.09

Nora, Dora and Jora each bought as many gobstoppers as they could with their money. They realised that they each then had exactly 9p left over. What is the greatest possible cost of a gobstopper?

p Correct Wrong

12. What is the smallest whole number that is exactly divisible by two thirds, four fifths and one half?

Correct Wrong

This is HCF and LCM level 7. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6


Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

Why am I learning this?

Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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Maths Mind Reader

Maths Mind Reader

This spectacular magic trick never fails to amaze people of all ages assuming they can add and subtract very simple numbers. The mathematics comes in finding out how the trick works.


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Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.


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When planning to use technology in your lesson always have a plan B!


Monday, May 13, 2024

"In my student days I worked my summers in America; at Camp Tamarack in New Jersey. I remember in 1979 being surrounded by large colourful insects called cicadas which can only be seen every 17 years.

In 2024, Illinois is experiencing a rare natural spectacle: the simultaneous emergence of cicada Broods XIII and XIX. This event, first recorded in 1803, won’t occur again until 2245. These periodical cicadas, which emerge either every 13 or 17 years, spend the majority of their lives underground, only surfacing to reproduce at the end of their long developmental cycles.

After mating, females lay approximately 500 to 600 eggs in woody plants before the adults die, their life above ground lasting merely a month. The eggs hatch six weeks later, and the nymphs burrow into the soil to begin their 13 or 17-year cycle.

This synchronisation of 13 and 17-year cycles is a perfect natural example of common multiples.


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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.

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Curriculum Reference

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



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:

Venn Diagram 1

Show the sets intersecting

Venn Diagram 2

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.

Indian Method


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.

HCF on a calculator

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.

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