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1.7 Prime Factors

GCSE Maths Foundation · Edexcel · Chapter 1: Number

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: prime factor trees, products of primes in index form, and HCF and LCM using prime factors. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Recognise prime numbers and explain why 1 is not prime
  • Write any number as a product of its prime factors using a factor tree or repeated division
  • Write a product of primes in index form
  • Use prime factors and Venn diagrams to find the HCF and LCM

Key words

  • Prime factor: A factor that is also a prime number. The prime factors of 12 are 2 and 3.
  • Product of primes: A number written as primes multiplied together: 12 = 2 × 2 × 3.
  • Index form: Repeated primes written as powers: 2 × 2 × 3 = 2² × 3.
  • Factor tree: A diagram that splits a number into factor pairs until every branch ends in a prime.

Prime numbers

A prime number has exactly two factors: 1 and itself. 1 is not prime (it has only one factor). 2 is the only even prime.

  • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Worked example: is it prime?

Is 91 prime?
Try dividing by the primes 2, 3, 5, 7: 91 ÷ 7 = 13.
So 91 = 7 × 13 is not prime.

Answer: no

Factor trees

Split the number into any factor pair. Keep splitting each branch until it ends in a prime. Circle the primes, then multiply them together, smallest first.

Worked example: product of primes

Write 180 as a product of its prime factors in index form.
180 = 2 × 2 × 3 × 3 × 5
Check: 4 × 9 × 5 = 180 ✓

Answer: 180 = 2² × 3² × 5

Key fact: Whichever factor pair you start with, you always end up with the same primes.

Repeated division

Another method is to keep dividing by the smallest prime that goes in exactly.

  • 2; Result 360 ÷ 2 = 180
  • 2; Result 180 ÷ 2 = 90
  • 2; Result 90 ÷ 2 = 45
  • 3; Result 45 ÷ 3 = 15
  • 3; Result 15 ÷ 3 = 5
  • 5; Result 5 ÷ 5 = 1

Worked example: index form

Read down the first column: 360 = 2 × 2 × 2 × 3 × 3 × 5

Answer: 360 = 2³ × 3² × 5

HCF and LCM using prime factors

Write both numbers as products of primes, then put the primes into a Venn diagram. Primes the numbers share go in the overlap.

Rule: HCF = multiply the numbers in the overlap. LCM = multiply every number in the diagram.

Worked example: HCF and LCM

60 = 2 × 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3
Shared: 2, 2 and 3. HCF = 2 × 2 × 3 = 12
LCM = 12 × 5 × 2 × 3 = 360

Answer: HCF 12, LCM 360

Worked example: square numbers

Show that 2⁴ × 3² is a square number.
2⁴ × 3² = (2² × 3) × (2² × 3) = 12 × 12 = 144
All the indices are even, so it is a square.

Answer: 144 = 12²

Common mistakes

  • Leaving a number that is not prime in the answer, e.g. 60 = 2² × 15. 15 must be split into 3 × 5.
  • Including 1 in a product of primes. 1 is not prime.
  • Adding the prime factors instead of multiplying them when you check.
  • Mixing up the HCF (overlap only) and the LCM (everything in the diagram).

Revision questions

Try these, then check your answers below each one.

Question 1. Which of these shows 60 written as a product of its prime factors? (Choose from: 2² × 3 × 5, 2 × 3 × 10, 4 × 15, 2² × 15.)

Answer: 2² × 3 × 5

Working: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. 10, 4 and 15 are not prime.

Question 2. Work out 2³ × 3 × 5.

Answer: 120

Working: 8 × 3 × 5 = 120.

Question 3. 360 = 2³ × 3² × 5 and 84 = 2² × 3 × 7. Find the HCF of 360 and 84.

Answer: 12

Working: Shared: 2² and 3. HCF = 2² × 3 = 12.

Related lessons

Continue in the full course

This lesson is section 1.7 (EGCMF0107) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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