Number
Grade 4-5HCF & LCM
HCF and LCM questions are easy to mix up, since both start from the same list of factors or multiples but ask for opposite things. This lesson covers finding the HCF and LCM by listing, using prime factorisation with a Venn diagram, and spotting which one a word problem actually needs.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the HCF by listing factors
The highest common factor (HCF) is the largest number that divides into two numbers exactly. List all the factors of each number, circle the factors they share, and pick the largest shared one.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest number in both lists is 12, so the HCF of 24 and 36 is 12.
Finding the LCM by listing multiples
The lowest common multiple (LCM) is the smallest number that both numbers divide into exactly. List the multiples of each number until a value appears in both lists, and pick the smallest one they share.
Multiples of 8: 8, 16, 24, ... Multiples of 12: 12, 24, ... The smallest number in both lists is 24, so the LCM of 8 and 12 is 24.
Using a Venn diagram of prime factors
Write each number as a product of prime factors, then place them in a Venn diagram, putting any primes shared by both numbers in the overlap. The HCF is the product of the primes in the overlap. The LCM is the product of every prime in the whole diagram, overlap included.
For 28 and 42: 28 = 2 × 2 × 7 and 42 = 2 × 3 × 7. The overlap contains 2 and 7, so the HCF = 2 × 7 = 14. The LCM uses every prime in the diagram: 2 × 2 × 7 × 3 = 84.
HCF and LCM in worded problems
A question needs the HCF when it asks for the greatest number of identical groups that can be made with nothing left over. A question needs the LCM when it asks when two repeating events will next happen at the same time.
"A shop has 48 pens and 60 pencils, packed into identical bags with none left over" needs the HCF. "Two bells ring every 8 seconds and every 12 seconds, when will they next ring together" needs the LCM.
Worked Examples
Three exam-style questions, fully solved.
Find the highest common factor (HCF) of 24 and 36.
Easy- 1.List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- 2.List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- 3.Find the largest number that appears in both lists
Answer: 12
A shop has 48 pens and 60 pencils. They are to be packed into identical bags, with no pens or pencils left over. What is the greatest number of bags that can be made?
Medium- 1.Recognise that "greatest number of identical groups with nothing left over" means finding the HCF of 48 and 60
- 2.Find the HCF of 48 and 60
Answer: 12 bags
The Venn diagram shows the prime factors of 28 and 42: 28 has 2 in its section, the overlap has 2 and 7, and 42 has 3 in its section. Find the HCF and LCM of 28 and 42.
Hard- 1.Find the HCF by multiplying the primes in the overlap: 2 × 7
- 2.HCF = 14
- 3.Find the LCM by multiplying every prime in the whole diagram: 2 (from 28) × 2 × 7 (overlap) × 3 (from 42)
Answer: HCF = 14, LCM = 84
Avoid These
The most common mistakes students make.
Confusing HCF and LCM, and finding the highest common factor when the question actually asks for the lowest common multiple, or the other way round.
Forgetting to include 1 and the number itself when listing factors, which can cause the HCF to be missed.
Stopping a list of multiples too early and missing the first value the two numbers share.
When using a Venn diagram, multiplying only the primes in the overlap to find the LCM, instead of multiplying every prime across the whole diagram.
Not recognising which operation a worded problem needs: "greatest number of identical groups with nothing left over" means HCF, while "when will they next happen together" means LCM.
FAQ
Questions parents and students ask.
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