Prime factors, HCF and LCM, explained
Every whole number greater than 1 can be built by multiplying prime numbers together, in exactly one way. Finding those prime factors is the quickest route to the highest common factor (HCF) and lowest common multiple (LCM) of two numbers.
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Prime numbers
A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Note that 1 is not prime, because it has only one factor, and 2 is the only even prime. Learning the primes up to 50 makes this whole topic faster, and spotting them quickly helps on the non-calculator paper.
Factor trees
To find the prime factors of a number, split it into any two factors, then keep splitting each factor until every branch ends in a prime. Circle the primes as you reach them. Whichever way you start, you always end with the same primes; this fact is called the Fundamental Theorem of Arithmetic.
Example 1: a factor tree
Write 84 as a product of its prime factors.
- 84 = 4 × 21 = 2 × 2 × 3 × 7.
- In index form: 84 = 2² × 3 × 7. Always check by multiplying back: 4 × 3 × 7 = 84.
Example 2: a larger number
Write 360 as a product of prime factors in index form.
- 360 = 36 × 10 = (4 × 9) × (2 × 5) = 2 × 2 × 3 × 3 × 2 × 5.
- 360 = 2³ × 3² × 5.
Repeated division
An alternative to a tree is to divide repeatedly by the smallest prime that goes in exactly. For 120: 120 ÷ 2 = 60, ÷ 2 = 30, ÷ 2 = 15, ÷ 3 = 5, ÷ 5 = 1. The divisors give 120 = 2³ × 3 × 5. This method is tidy and hard to get wrong.
Highest common factor (HCF)
The HCF of two numbers is the largest number that divides into both. For small numbers, list the factors of each and pick the biggest common one. For larger numbers, use prime factors: the HCF is the product of the primes they share, using the lowest power of each shared prime.
Lowest common multiple (LCM)
The LCM is the smallest number that both divide into. For small numbers, list multiples until one appears in both lists. For larger numbers, use prime factors: the LCM is the product of every prime that appears in either number, using the highest power of each.
The Venn diagram method
Draw two overlapping circles. Put the prime factors the numbers share in the overlap, and the remaining prime factors of each number in its own section. Then:
- HCF = the product of the numbers in the overlap.
- LCM = the product of all the numbers in the diagram.
Worked examples
Example 3: HCF and LCM by prime factors
Find the HCF and LCM of 84 and 120.
- 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
- HCF: shared primes with the lowest powers: 2² × 3 = 12.
- LCM: all primes with the highest powers: 2³ × 3 × 5 × 7 = 840.
Example 4: the Venn method
Find the HCF and LCM of 36 and 60.
- 36 = 2 × 2 × 3 × 3 and 60 = 2 × 2 × 3 × 5.
- Overlap: 2, 2, 3. 36 only: 3. 60 only: 5.
- HCF = 2 × 2 × 3 = 12. LCM = 3 × 2 × 2 × 3 × 5 = 180.
Example 5: listing for small numbers
Find the LCM of 12 and 18.
- Multiples of 12: 12, 24, 36, 48, … Multiples of 18: 18, 36, …
- LCM = 36.
A useful check
For any two numbers, HCF × LCM = the product of the two numbers. In Example 3, 12 × 840 = 10,080 = 84 × 120. If this check fails, recheck your prime factors before going any further.
Word problems: is it HCF or LCM?
The trickiest part of exam questions is deciding which one is needed.
- HCF questions are about splitting things into equal groups as large as possible: the biggest identical packs, the largest square tiles that fit exactly, the most bags with no leftovers.
- LCM questions are about things that repeat and happen together again: buses leaving at the same time, lights flashing together, the smallest number of items that can be packed in either size of box.
Example 6: an HCF problem
A teacher has 48 pencils and 72 rubbers. She wants to make identical packs with no items left over. What is the greatest number of packs?
- Identical packs, as many as possible, with nothing left: HCF of 48 and 72.
- 48 = 24 × 3 and 72 = 2³ × 3², so HCF = 2³ × 3 = 24 packs, each with 2 pencils and 3 rubbers.
Example 7: an LCM problem
Three lights flash every 6, 8 and 10 seconds. They flash together now. When will they next flash together?
- Repeating events happening together: LCM of 6, 8 and 10.
- 6 = 2 × 3, 8 = 2³, 10 = 2 × 5. LCM = 2³ × 3 × 5 = 120 seconds, which is 2 minutes.
Common mistakes
- Including 1 as a prime factor.
- Stopping a factor tree before every branch ends in a prime.
- Mixing up the rules: lowest powers for HCF, highest powers for LCM.
- Choosing HCF when the question is about events repeating, or LCM when it is about splitting into groups.
Practice questions
Try each one before opening the answer.
Q1. Write 90 as a product of prime factors in index form.
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Q2. Find the HCF of 24 and 40.
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Q3. Find the LCM of 15 and 20.
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Q4. A = 2³ × 5² and B = 2² × 5 × 7. Find the HCF and LCM.
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Frequently asked questions
Is this topic on the Foundation paper?
Yes. Primes, factors, multiples, HCF and LCM are on both tiers, and prime factor decomposition in index form is common on non-calculator papers.
Is 1 a prime number?
No. A prime has exactly two different factors, 1 and itself. The number 1 has only one factor, so it is not prime.
Key takeaways
- Primes have exactly two factors; 1 is not prime and 2 is the only even prime.
- Use a factor tree or repeated division to find prime factors.
- HCF: shared primes, lowest powers. LCM: all primes, highest powers.
- In a Venn diagram, HCF is the overlap and LCM is everything.
- Groups suggest HCF; repeating events suggest LCM.