GCSE Revision

Prime factors, HCF and LCM

Prime factor trees

Every whole number bigger than 1 can be written as a product of in exactly one way.

  1. Split the number into any factor pair.
  2. Circle any prime; keep splitting the others.
  3. Stop when every branch ends in a prime.
  4. Write the primes multiplied together, smallest first, in .

Example: write 360 as a product of prime factors.

  1. 360 = 36 × 10
  2. 36 = 4 × 9 = 2 × 2 × 3 × 3
  3. 10 = 2 × 5
  4. 360 = 2 × 2 × 2 × 3 × 3 × 5
  5. 360 = 2³ × 3² × 5

Check: 8 × 9 × 5 = 360.

Exam tip

"Write as a product of prime factors" needs × signs. A list such as 2, 3, 5 doesn't get the mark.

Key terms

Highest common factor (HCF)

The (HCF) of two numbers is the biggest number that divides into both.

  1. Write each number as a product of prime factors.
  2. Pick out the primes that are in both.
  3. Use the lower power of each and multiply.

For small numbers you can list the factors of each instead and pick the biggest in both lists.

Example: find the HCF of 360 and 84.

  1. 360 = 2³ × 3² × 5
  2. 84 = 2² × 3 × 7
  3. Primes in both: 2 and 3
  4. Lower powers: 2² and 3
  5. HCF = 2² × 3 = 4 × 3 = 12

Check: 360 ÷ 12 = 30 and 84 ÷ 12 = 7, and 30 and 7 have no common factor bigger than 1.

Key terms

Lowest common multiple (LCM)

The (LCM) of two numbers is the smallest number that both divide into.

  1. Write each number as a product of prime factors.
  2. Take every prime that appears in either number.
  3. Use the higher power of each and multiply.

Example: find the LCM of 360 and 84.

  1. 360 = 2³ × 3² × 5
  2. 84 = 2² × 3 × 7
  3. Higher powers: 2³, 3², 5 and 7
  4. LCM = 8 × 9 × 5 × 7 = 2520

Harder example: buses leave every 12 minutes and every 18 minutes. Both leave at 9:00 am. When do they next leave together?

  1. 12 = 2² × 3 and 18 = 2 × 3²
  2. LCM = 2² × 3² = 36 minutes
  3. 9:36 am

Remember

"Next together?" needs the LCM. "Largest equal groups?" needs the HCF.

Key terms

Using a Venn diagram

A Venn diagram of the prime factors gives the HCF and LCM together.

  1. Draw two overlapping circles, one per number.
  2. Put the shared primes in the overlap, and each number's other primes in its own circle.
  3. HCF = the product of the overlap.
  4. LCM = the product of everything in the diagram.

Example: find the HCF and LCM of 360 and 84 with a Venn diagram.

  1. 360 = 2 × 2 × 2 × 3 × 3 × 5 and 84 = 2 × 2 × 3 × 7
  2. Overlap: 2, 2, 3. Only 360: 2, 3, 5. Only 84: 7
  3. HCF = 2 × 2 × 3 = 12
  4. LCM = 12 × (2 × 3 × 5) × 7 = 12 × 30 × 7 = 2520

Take the quiz