What are HCF and LCM?
HCF and LCM both compare two (or more) numbers, but in opposite directions:
- HCF (Highest Common Factor) — the largest number that divides exactly into both numbers
- LCM (Lowest Common Multiple) — the smallest number that both numbers divide exactly into
Both can be found by listing factors or multiples for small numbers, but for larger numbers the reliable method is prime factorisation — see the Factors, Multiples and Primes page for how to build a factor tree.
HCF and LCM by listing
Worked Example 1 — HCF by listing
Find the HCF of 18 and 24.
1
List the factors of 18: 1, 2, 3, 6, 9, 18
2
List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
3
Find the factors common to both lists: 1, 2, 3, 6 — the highest is 6
AnswerHCF(18, 24) = 6
Worked Example 2 — LCM by listing
Find the LCM of 4 and 6.
1
List multiples of 4: 4, 8, 12, 16, 20, 24…
2
List multiples of 6: 6, 12, 18, 24…
3
Find the smallest number in both lists: 12
AnswerLCM(4, 6) = 12
HCF and LCM using prime factorisation
Listing works well for small numbers, but is slow and error-prone for larger ones. Instead, write both numbers as products of their prime factors, then compare.
Worked Example 3
Find the HCF and LCM of 36 and 60 using prime factorisation.
1
Write each number as a product of primes: 36 = 2² × 3², and 60 = 2² × 3 × 5
2
HCF: take the lowest power of each prime that appears in both lists: 2² and 3¹ (5 doesn't appear in 36, so it's excluded) → 2² × 3 = 12
3
LCM: take the highest power of each prime that appears in either list: 2², 3², 5¹ → 2² × 3² × 5 = 180
AnswerHCF = 12, LCM = 180
The overlapping region holds the shared prime factors (used for the HCF); everything across both circles is used for the LCM.
HCF and LCM in context
Worked Example 4
Two bus routes leave the same station. Route A departs every 15 minutes, Route B departs every 20 minutes. If they both leave together at 9:00am, when will they next leave together?
1
This is an LCM problem — find the smallest time that is a multiple of both 15 and 20
2
15 = 3 × 5, 20 = 2² × 5. LCM = 2² × 3 × 5 = 60
3
Add 60 minutes to 9:00am
Answer10:00am
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Find the HCF of 12 and 18 by listing factors.Foundation
Q2Find the LCM of 3 and 5 by listing multiples.Foundation
Q3Find the HCF of 20 and 30.Foundation
Q4Find the LCM of 6 and 9.Foundation
Q5Two lighthouses flash every 8 seconds and every 12 seconds. If they flash together at the same moment, how many seconds until they next flash together?Foundation
Higher (Grade 5–7)
Q6Given that 48 = 2⁴ × 3 and 180 = 2² × 3² × 5, find the HCF of 48 and 180.Higher
Q7Given that 48 = 2⁴ × 3 and 180 = 2² × 3² × 5, find the LCM of 48 and 180.Higher
Q8Find the HCF and LCM of 42 and 70 using prime factorisation.Higher
Q9A shop packs pens into boxes of 24 and pencils into boxes of 18. What is the smallest number of pens and pencils that can be packed so both amounts use whole boxes?Higher
Q10Two ribbon lengths of 84cm and 126cm are cut into equal pieces with no ribbon left over. Find the greatest possible length of each piece.Higher
Higher — Hard (Grade 8–9)
Q11Two numbers have HCF 6 and LCM 180. One of the numbers is 36. Find the other number.Grade 8–9
Q12Prove that for any two numbers a and b, HCF(a, b) × LCM(a, b) = a × b, and use this to find the LCM of 28 and 42 given that their HCF is 14.Grade 8–9
Q13Find the HCF and LCM of 2³ × 3² × 5 and 2² × 3³ × 7.Grade 8–9
Q14Three bells ring every 6, 8 and 10 seconds. If they ring together at time zero, find the next time all three ring together.Grade 8–9
Q15Two numbers are both multiples of 12 and both factors of 240. List all the possible pairs, given the numbers must be different.Grade 8–9
Answers
Foundation (Q1–Q5)
Q16
Q215
Q310
Q418
Q524 seconds(LCM of 8 and 12)
Higher (Q6–Q10)
Q612(2² × 3)
Q7720(2⁴ × 3² × 5)
Q8HCF = 14, LCM = 210(42 = 2×3×7, 70 = 2×5×7)
Q972 of each(LCM of 24 and 18)
Q1042cm(HCF of 84 and 126)
Higher — Hard (Q11–Q15)
Q1130(6 × 180 ÷ 36 = 30)
Q12LCM = 84(28 × 42 ÷ 14 = 84)
Q13HCF = 2² × 3² = 36; LCM = 2³ × 3³ × 5 × 7 = 7560
Q14120 seconds(LCM of 6, 8 and 10 = 2³ × 3 × 5)
Q15(12, 24), (12, 60), (12, 120), (12, 240), (24, 60), (24, 120), (24, 240), (60, 120), (60, 240), (120, 240)(multiples of 12 that also divide 240: 12, 24, 60, 120, 240)
Common mistakes
Common Mistake 1
Mixing up HCF and LCM
HCF is always smaller than or equal to both starting numbers; LCM is always bigger than or equal to both. If your HCF answer is larger than one of the original numbers, you've made an error.
Common Mistake 2
Using the wrong power when combining prime factorisations
For the HCF, take the lowest power of each shared prime. For the LCM, take the highest power of each prime appearing in either number. Mixing these up is the most common error in this topic.
Common Mistake 3
Forgetting a prime that only appears in one number
For the LCM, every prime that appears in either number must be included — even if it only appears in one of them. Leaving one out gives an LCM that isn't actually a multiple of both numbers.
Common Mistake 4
Not recognising "when will they next happen together" as an LCM question
Real-world questions about repeating events (buses, bells, flashing lights) are LCM problems in disguise — look for the phrase "at the same time" or "together" as the signal.
Exam tips
💡 Exam Tip 1
Use prime factorisation for anything beyond small numbers
Listing factors or multiples is fine for numbers under about 30, but becomes slow and error-prone for larger numbers — switch to prime factorisation as soon as the numbers get bigger.
💡 Exam Tip 2
Write both factorisations with all primes listed, using power 0 where needed
Lining up 36 = 2² × 3² × 5⁰ and 60 = 2² × 3¹ × 5¹ side by side makes it much easier to spot the lowest and highest powers without missing anything.
💡 Exam Tip 3
Check your HCF divides both numbers exactly
Once you have an answer, do a quick check: does your HCF divide exactly into both original numbers? Does each original number divide exactly into your LCM? This catches most errors instantly.
💡 Exam Tip 4
Remember the shortcut HCF × LCM = a × b
For two numbers a and b, HCF(a,b) × LCM(a,b) always equals a × b. This is a useful check, and sometimes lets you find one value quickly if you already know the other three.
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