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Highest Common Factor and LCM: Free Printable Worksheet for Year 7
- ForYear 7
- StageKey Stage 3
- CostFree
- FormatPrintable PDF
Highest Common Factor and LCM is a free printable maths worksheet for Year 7 (ages 11 to 12) in Key Stage 3. Children work through seven questions on the highest common factor (HCF) and the lowest common multiple (LCM) of pairs of numbers, using prime factors. It takes about 15 minutes and needs only a pencil.
It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages. Give page 1 to the child and keep page 2, which has the notes and answers. Black and white printing works well.
What is on the sheet
The sheet has seven questions with space to answer under each one. Children find the highest common factor of some pairs of numbers and the lowest common multiple of others. The numbers are small enough to work with by hand, so the focus stays on method. Children can list factors or multiples, or break each number into prime factors, and a small Venn diagram in the margin helps them organise their thinking.
- HCF of 12 and 18
- HCF of 20 and 30
- HCF of 24 and 36
- LCM of 4 and 6
- LCM of 5 and 8
- LCM of 12 and 15
A worked example with different numbers
Find the HCF and the LCM of 14 and 21.
This pair is not on the sheet. It shows two ways to work, so children can choose the one that suits them.
Method 1: list factors and multiples- Factors of 14: 1, 2, 7, 14. Factors of 21: 1, 3, 7, 21.
- The biggest factor in both lists is 7. That is the HCF.
- Multiples of 14: 14, 28, 42. Multiples of 21: 21, 42.
- The first number in both lists is 42. That is the LCM.
- Write each number as a product of primes: 14 = 2 x 7 and 21 = 3 x 7.
- For the HCF, multiply the primes that appear in both. Only 7 is shared, so the HCF is 7.
- For the LCM, multiply every prime needed to build both numbers, counting the shared 7 once: 2 x 3 x 7 = 42.
- Check: 7 x 42 = 294 and 14 x 21 = 294.
A method that works every time
Break each number into primes
Write each number as a product of prime numbers. A factor tree helps. Keep splitting until every branch ends in a prime.
Line up the matching primes
Write the two lists one above the other. Circle or tick the primes that appear in both. A Venn diagram shows this well, with shared primes in the middle.
Build the HCF
Multiply the primes in the shared part. If a prime is shared more than once, include it each time it is shared.
Build the LCM
Multiply every prime in the whole diagram, with the shared ones counted once. Then check that the answer is a multiple of both starting numbers.
Why factors and multiples matter
Factors and multiples are two sides of the same idea. A factor divides exactly into a number. A multiple is what you get when you multiply a number by a whole number. The highest common factor looks down at the shared factors of two numbers. The lowest common multiple looks up at the shared multiples. Seeing them as a pair makes it easier to remember which is which.
Prime factors give a reliable way to find both answers, even when the numbers get larger and listing becomes slow. Every whole number above 1 can be written as a product of primes in only one way, apart from the order. That is why comparing two prime factor lists works. These skills are used later when simplifying fractions, adding fractions with different denominators and solving problems about repeating patterns.
Skills and curriculum link
- Factors and multiples
- Prime factors
- Highest common factor
- Lowest common multiple
- Using Venn diagrams
- Checking answers
Ideas that help with HCF and LCM
- Factors come in pairsIf 3 times 5 makes 15, then 3 and 5 are both factors of 15. Finding one factor often gives you its partner, so lists are quicker to build.
- The HCF is never bigger than the smaller numberA factor of a number cannot be larger than the number itself. This is a quick way to spot a mistake.
- The LCM is never smaller than the larger numberA multiple of a number is at least as big as the number. If an LCM is smaller than one of the starting numbers, something has gone wrong.
- Primes are the building blocksEvery whole number above 1 is either prime or can be built by multiplying primes. Splitting numbers into primes makes shared parts easy to see.
Key words
- FactorA whole number that divides exactly into another number.
- MultipleA number you get by multiplying a whole number by another whole number.
- Prime numberA whole number above 1 with exactly two factors: 1 and itself.
- Prime factorA factor that is also a prime number.
- CommonShared by two or more numbers. A common factor is a factor of both.
- Venn diagramA diagram with overlapping circles. The overlap shows what the two groups share.
How to use it
- Print both pages. Page 1 is for the child. Keep page 2, which has the notes and answers.
- Read the first question together. Ask whether it wants a highest common factor or a lowest common multiple, and what that means.
- Let the child choose a method: listing, prime factors or a small Venn diagram in the margin. Allow about 15 minutes.
- Check against the answers on page 2. Ask, "How did you know?" The method matters as much as the answer.
- Come back another day and redo any questions that felt tricky. Repeating a familiar sheet builds confidence.
Help, stretch and game ideas
If it feels too hard
List all the factors of each number first, then circle the biggest one they share. For the lowest common multiple, write out the first few multiples of each number and look for the first one that appears in both lists. Try the first questions before moving on.
To stretch confident children
Find the HCF and LCM of 36 and 48, then check that the HCF multiplied by the LCM equals 36 x 48. Then try your own pair of numbers. Does the same check work every time? Explain why you think so.
Make it a game
Race a partner. Pick a pair of numbers. One player finds the HCF and the other finds the LCM. Check each other's work, score a point for each correct answer, then swap jobs and choose a new pair.
Mistakes to watch for
Watch forA child multiplies the two numbers together and calls that the LCM. For 10 and 14 they write 140.
Try thisAsk whether a smaller number appears in both lists of multiples. List multiples of each and look for the first match, which here is 70.
Watch forA child confuses the two terms and gives a number bigger than both starting numbers as the HCF.
Try thisRemind them that a factor divides into a number, so it cannot be bigger than it. Ask, "Can your answer divide into both numbers?"
Watch forWith 16 and 24, a child circles only one shared 2 in the prime factors and writes 2 as the HCF.
Try thisWrite each prime factor list in full and match them one for one. Count how many 2s both lists share, then multiply the shared primes.
Talk about it
- What is the difference between a factor and a multiple? Can you give an example of each?
- How do you know your answer is a factor of both numbers?
- Why must the LCM be at least as big as the larger number?
- Which method did you prefer, listing or prime factors? When might the other one be better?
- What does the overlap of your Venn diagram show?
- Can you think of two numbers whose HCF is 1? What do you notice about them?
Take it off the page
Factor hunt with counters
Take a number of counters, such as 16, and arrange them into as many different rectangles as you can. Each rectangle shows a pair of factors. Write the pairs down.
Factor tree art
Draw a factor tree for a number of your choice. Colour each prime a different colour. Do the same for a second number and compare which colours match.
Clap and stamp
Pick two numbers. One person claps on every multiple of the first, the other stamps on every multiple of the second. Count how far you go before you both move at once.
Who it is for
Parents
Sit beside your child for the first question, then let them lead. Ask them to explain what each term means before they start. Keep sessions short and praise clear thinking, not just correct answers.
Teachers
Use it as a starter, plenary, early finisher task or homework after teaching factors, multiples and prime factorisation. It takes about 15 minutes. Ask pupils to show a Venn diagram, then compare methods in pairs.
Home educators
Print it as part of a wider maths plan on number. It works for revision of factors and multiples and can be reprinted for brothers and sisters. Ask the learner to teach you one method.
Where this fits in maths
Before this, children learn multiplication facts, find factors and multiples of single numbers, and meet prime numbers. They also practise division, because checking a factor means checking whether a number divides exactly.
After this, the same ideas help with simplifying fractions, adding fractions with different denominators, and solving problems about repeating events. In Key Stage 3, children also use prime factorisation more widely, and later meet ideas such as powers and indices.
Questions parents and teachers ask
- What age is Highest Common Factor and LCM for?
It is written for Year 7 (ages 11 to 12) in Key Stage 3. It can also work as revision for older children, or as a stretch task for confident younger learners who already know their times tables.
- What will children practise?
They practise finding the highest common factor and the lowest common multiple of pairs of numbers, using prime factors. They also use factors, multiples and a small Venn diagram to organise their thinking.
- How long does it take?
It takes about 15 minutes. Children who like to draw a Venn diagram for every question may take a little longer. Work through the first question together, then let them continue on their own.
- What if my child finds it hard?
Start with listing. Ask them to write all the factors of each number and circle the biggest one they share. For multiples, list a few of each and look for the first match. Page 2 has more help.
- Are the answers included?
Yes. Page 2 is for the grown-up and has the answers, notes on how to use the sheet, help for children who find it hard and a stretch idea. Keep it out of sight until the child has finished.
- Is it free, and do I need colour printing?
Yes, it is free to print for home and classroom use. You do not need colour. It prints well in black and white on plain A4 paper, and a pencil is all the child needs.
- Can teachers and home educators use it?
Yes. Teachers can use it as a starter, plenary, early finisher task, homework or table activity with no preparation beyond printing. Home educators and tutors can use it alone or as part of a wider plan, and reprint it for revision.
Preview the full PDF
Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.