Algebra · GCSE Maths
HCF and LCM
GCSE Maths HCF and LCM: prime factor trees, index form, the Venn method for two numbers, and HCF × LCM = product for a pair.
HCF is the highest common factor — the biggest number that divides both. LCM is the lowest common multiple — the smallest number both divide into. Prime factorise first.
The important bits
What you need to know
- 1
A factor divides exactly. A multiple is in the times table. 6 is a factor of 24; 24 is a multiple of 6.
- 2
The HCF (highest common factor) is the largest number that divides both integers exactly. HCF of 24 and 36 is 12.
- 3
The LCM (lowest common multiple) is the smallest positive integer that is a multiple of both. LCM of 4 and 6 is 12.
- 4
Prime factor trees write a number as a product of primes: 36 = 2² × 3², 60 = 2² × 3 × 5.
- 5
HCF from prime form: take each common prime with the lowest index. LCM: take each prime that appears with the highest index.
- 6
Venn diagram for two numbers: overlap is HCF primes; union is LCM. HCF × LCM = product of the two original numbers.
- 7
For three numbers, HCF is the product of common primes with lowest indices across all three; LCM is more careful — list every prime with its highest index from any number.
- 8
Word problems: “bells ring together” and “tiles fit exactly” are LCM. “Cut into equal pieces” and “identical groups” are HCF.
Quotations worth analysing
Short evidence. Real method.
“HCF × LCM = a × b”
Only for two numbers at GCSE. If HCF is 6 and one number is 36, the other is 36 × LCM ÷ 6. Misapplying to three numbers breaks the identity.
“Lowest index for HCF, highest index for LCM”
HCF uses the overlap. LCM uses the full spread. Swapping lowest and highest is the standard error on 2² × 3 versus 2³ × 3².
“12 = 2² × 3”
Writing 12 = 2 × 2 × 3 as 2² × 3 makes HCF and LCM mechanical. Trees without index form still work but invite missed factors.
Go deeper
Prime factor trees without lost primes
Break 60: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Break 84: 84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7. HCF: common primes are 2 and 3. Lowest powers: 2² and 3¹, so HCF = 4 × 3 = 12. LCM: all primes from either number with highest powers: 2², 3¹, 5¹, 7¹, so LCM = 4 × 3 × 5 × 7 = 420. Check: 12 × 420 = 5040 and 60 × 84 = 5040. Listing multiples until a match works for small numbers but fails on 60 and 84 before the list gets long. Index form is the scalable method.
Go deeper
The Venn picture for two numbers
Write prime factors of 60 and 84 in a two-circle Venn. Shared factors in the overlap: 2² and 3. Left only: 5. Right only: 7. HCF is the product of the overlap: 12. LCM is the product of everything in the diagram: 2² × 3 × 5 × 7 = 420. Students who put all of 60 in one circle and all of 84 in the other without splitting shared primes double-count in the LCM or under-count the HCF. Split each number into shared and exclusive parts first. For HCF of 18 and 24: 18 = 2 × 3², 24 = 2³ × 3. HCF = 2 × 3 = 6. LCM = 2³ × 3² = 72. The Venn is a picture of the index rule, not a separate trick.
Go deeper
Word problems: together versus equal pieces
LCM: “Bus A every 8 minutes, Bus B every 12 minutes — together again?” Times align at LCM(8, 12) = 24 minutes. “Square tiles on a 60 cm by 84 cm floor with no cuts” needs the largest square tile: HCF(60, 84) = 12 cm tiles. HCF questions ask for the largest equal grouping or largest dimension that fits both. LCM questions ask for the first time events coincide or the smallest container that fits both repeat cycles. If the question says “lowest” and “both,” think LCM. If it says “highest” and “divides both,” think HCF. Three-number LCM: 4 = 2², 6 = 2 × 3, 10 = 2 × 5, LCM = 2² × 3 × 5 = 60. Three-number HCF: only primes common to all three with lowest index.
See the idea in action
Find the HCF and LCM of 60 and 84. Step 1: 60 = 2² × 3 × 5. 84 = 2² × 3 × 7. Step 2: HCF = 2² × 3 = 12 (common primes, lowest indices). Step 3: LCM = 2² × 3 × 5 × 7 = 420 (all primes, highest indices). Step 4: Check HCF × LCM = 12 × 420 = 5040. 60 × 84 = 5040. Application: largest square tile on a 60 cm by 84 cm rectangle — tile side 12 cm (HCF).
Exam technique
Turn knowledge into marks
Write both numbers in index form before you combine. Circle common primes for HCF; underline highest powers for LCM.
Common mistakes
Do not give these marks away
- 01
Using highest indices for HCF or lowest indices for LCM.
- 02
Multiplying the two numbers to find LCM instead of using prime structure.
- 03
Applying HCF × LCM = product to three numbers, where the identity does not hold.
HCF of 24 and 36 is 12. LCM of 24 and 36 is
A72
B864
C12
D24
Show the answer
72. 24 = 2³ × 3, 36 = 2² × 3². LCM = 2³ × 3² = 8 × 9 = 72. 864 is 24 × 36. 12 is the HCF. 24 divides 36 but is not the LCM.
Quick questions
If this is the bit you searched
What is the difference between HCF and LCM?
HCF is the largest number that divides both exactly. LCM is the smallest number both divide into. One is about factors; the other about multiples.
Can I list multiples instead of factor trees?
For small numbers, yes. For 60 and 84, listing is slow. Prime factors scale and earn method marks.
Does HCF × LCM always equal the product?
For two positive integers, yes. For three or more numbers, use prime methods directly; do not rely on that product rule.
What if one number is a factor of the other?
The smaller number is the HCF. The larger is the LCM. Example: HCF(6, 18) = 6, LCM(6, 18) = 18.