Prime Factors, HCF and LCM
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Revision notes for Edexcel GCSE Maths Prime Factors, HCF and LCM. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Prime Factors, HCF and LCM

What you'll learn

  • How to recognise factors, multiples and prime numbers.
  • How to write a number as a product of prime factors.
  • How to find the HCF and LCM of two or more numbers.
  • How to use LCM for “next time together” problems, such as buses or flashing lights.

Factors, multiples and primes

Before HCF and LCM, you need three basic ideas.

Definition

Key number words

  • A factor of a number divides it exactly, with no remainder.
  • A multiple of a number is in its times table.
  • A prime number is a whole number greater than 1 with exactly two factors: 1 and itself.
  • A composite number is a whole number greater than 1 that is not prime.
Example

Checking factors and multiples

  1. To check if 6 is a factor of 42, divide 42 by 6.

An array model shows that 42 splits exactly into 6 equal groups of 7, so 6 is a factor of 42.

  1. 42 divided by 6 gives 7 exactly, so 6 is a factor of 42.

  2. To find multiples of 7, count in 7s.

  3. The first few multiples of 7 are 7, 14, 21, 28 and 35.

Common Mistake

1 is not prime

1 only has one factor, so it is not prime. Your prime factorisation should never include 1.

Writing numbers as products of prime factors

A product means a multiplication. So a product of prime factors means “prime numbers multiplied together”.

Definition

Prime factorisation

A prime factorisation writes a number as a multiplication of prime numbers only. Index notation uses powers, for example 232^323 means 2×2×22 \times 2 \times 22×2×2.

You can use a factor tree, or keep dividing by small prime numbers.

Example

Writing a number as prime factors

  1. Start with 84 and split it into factors: 84=2×4284 = 2 \times 4284=2×42.

A factor tree breaks 84 down into the prime factors 2, 2, 3 and 7.

  1. Keep splitting any number that is not prime: 42=2×2142 = 2 \times 2142=2×21 and 21=3×721 = 3 \times 721=3×7.

  2. Collect only the prime numbers: 2, 2, 3 and 7.

  3. Write the answer using powers: 84=22×3×784 = 2^2 \times 3 \times 784=22×3×7.

Tip

Use small primes first

Try dividing by 2, then 3, then 5, then 7. This keeps your working organised and helps you avoid missing factors.

Highest Common Factor, HCF

Definition

Highest Common Factor

The highest common factor, or HCF, is the largest number that is a factor of all the numbers you are comparing.

To find an HCF using prime factors:

  • write each number as prime factors
  • keep only the primes that appear in every number
  • choose the smallest power of each shared prime
Example

Finding an HCF

  1. Find the HCF of 54 and 126.

A prime-factor comparison shows that the HCF uses only the shared prime factors with the lowest powers.

  1. Write each number as prime factors: 54=2×3354 = 2 \times 3^354=2×33.

  2. Factorise the other number: 126=2×32×7126 = 2 \times 3^2 \times 7126=2×32×7.

  3. The common primes are 2 and 3.

  4. Choose the smallest powers: 2 and 323^232.

  5. Multiply them: 2×32=182 \times 3^2 = 182×32=18, so the HCF is 18.

Key Idea

HCF method

For HCF, keep only the prime factors that appear in every number, and use the smallest power of each.

Lowest Common Multiple, LCM

Definition

Lowest Common Multiple

The lowest common multiple, or LCM, is the smallest positive number that is a multiple of all the numbers you are comparing.

To find an LCM using prime factors:

  • write each number as prime factors
  • include every prime that appears
  • choose the largest power of each prime
Example

Finding an LCM

  1. Find the LCM of 45 and 60.

A prime-factor comparison shows that the LCM uses every prime factor needed, taking the highest powers.

  1. Write 45 as prime factors: 45=32×545 = 3^2 \times 545=32×5.

  2. Write 60 as prime factors: 60=22×3×560 = 2^2 \times 3 \times 560=22×3×5.

  3. Include every prime that appears: 2, 3 and 5.

  4. Choose the largest powers: 222^222, 323^232 and 5.

  5. Multiply them: 22×32×5=1802^2 \times 3^2 \times 5 = 18022×32×5=180, so the LCM is 180.

Key Idea

HCF vs LCM

HCF uses common primes with lowest powers. LCM uses all primes with highest powers.

Common Mistake

Mixing up HCF and LCM

If your HCF is bigger than one of the original numbers, something has gone wrong. If your LCM is smaller than one of the original numbers, something has gone wrong.

Repeated events: buses, lights and bells

When two things repeat regularly, the next time they happen together is found using the LCM of their time intervals.

Example

Finding the next time together

  1. A tram arrives every 9 minutes. A coach arrives every 12 minutes. Both arrive at 10:00.

Two timelines show the first shared arrival after 10:00 occurs when the 9-minute and 12-minute patterns meet.

  1. Find the LCM of 9 and 12.

  2. Factorise: 9=329 = 3^29=32 and 12=22×312 = 2^2 \times 312=22×3.

  3. Use the highest powers: 22×32=362^2 \times 3^2 = 3622×32=36.

  4. They next arrive together 36 minutes after 10:00, which is 10:36.

Common Mistake

Use the same units

Before finding an LCM in a time question, make sure all intervals are in the same units, such as all seconds or all minutes.

When prime factors are already given

Sometimes the numbers are already written in prime factor form. This is good news: you do not need to make a factor tree again.

Example

Using prime powers already given

Suppose

P=23×32×5P = 2^3 \times 3^2 \times 5P=23×32×5

and

Q=22×3×7![Aprime−powertablemakesitclearwhichpowersarechosenfortheHCFandfortheLCM.](https://assets.mathsgenie.co.uk/notes/diagrams/3695d5de−aa06−4682−abf0−8f1cc4f1544a.png)Q = 2^2 \times 3 \times 7 ![A prime-power table makes it clear which powers are chosen for the HCF and for the LCM.](https://assets.mathsgenie.co.uk/notes/diagrams/3695d5de-aa06-4682-abf0-8f1cc4f1544a.png)Q=22×3×7![Aprime−powertablemakesitclearwhichpowersarechosenfortheHCFandfortheLCM.](https://assets.mathsgenie.co.uk/notes/diagrams/3695d5de−aa06−4682−abf0−8f1cc4f1544a.png)

  1. For the HCF, use only primes in both expressions: 2 and 3.

  2. Choose the lower powers: 222^222 and 3, so HCF=22×3=12\text{HCF} = 2^2 \times 3 = 12HCF=22×3=12.

  3. For the LCM, include every prime that appears: 2, 3, 5 and 7.

  4. Choose the higher powers: 232^323, 323^232, 5 and 7, so LCM=23×32×5×7=2520\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 2520LCM=23×32×5×7=2520.

HCF and LCM of three numbers

The same rules work for three or more numbers.

For HCF, a prime must appear in all the numbers.
For LCM, a prime only needs to appear in at least one number.

Example

HCF and LCM of three numbers

  1. Find the HCF and LCM of 12, 18 and 30.

A three-number prime-factor table shows that the HCF needs primes in every number, while the LCM uses all primes present.

  1. Factorise each number: 12=22×312 = 2^2 \times 312=22×3, 18=2×3218 = 2 \times 3^218=2×32, and 30=2×3×530 = 2 \times 3 \times 530=2×3×5.

  2. For the HCF, keep primes in all three and choose the lowest powers: 2×3=62 \times 3 = 62×3=6.

  3. For the LCM, include all primes and choose the highest powers: 22×32×5=1802^2 \times 3^2 \times 5 = 18022×32×5=180.

  4. The HCF is 6 and the LCM is 180.

Finding two numbers from their HCF and LCM

This is a trickier style of question, but you can work logically.

Definition

Coprime

Two numbers are coprime if their HCF is 1. For example, 3 and 4 are coprime.

If the HCF is 6, both numbers must be multiples of 6. Once you take out that 6, the leftover parts must be coprime.

Example

Finding the two numbers

  1. Two whole numbers greater than 8 have HCF 6 and LCM 72.

A factor-pair diagram shows that after removing the HCF, the remaining coprime parts must multiply to 12.

  1. Since the HCF is 6, both numbers are multiples of 6.

  2. Divide the LCM by the HCF: 72 divided by 6 is 12. The leftover parts must multiply to 12.

  3. Factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4.

  4. The leftover pair must be coprime. 2 and 6 are not coprime.

  5. 1 and 12 would give 6 and 72, but 6 is not greater than 8.

  6. Use 3 and 4, giving the numbers 18 and 24.

  7. Check: the HCF of 18 and 24 is 6, and the LCM is 72.

Exam technique

In the exam

  1. Write prime factorisations clearly, using powers if a factor repeats.

  2. For HCF, ask: “What do all the numbers share?”

  3. For LCM or repeated events, ask: “When is the first time they meet again?”

  4. Check the size: HCF should not be bigger than the smallest number, and LCM should not be smaller than the largest.

Self review

Check yourself

  • Can you write 96 as a product of prime factors?

  • If two lights flash every 6 seconds and 15 seconds, what calculation finds when they next flash together?

  • In prime factor form, do you use the lowest powers or highest powers for the HCF?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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Prime Factors, HCF and LCM Revision Guide

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