Prime Factor Decomposition (AQA GCSE Maths): Revision Note

Prime factor decomposition

What are prime factors?

• A factor of a given number is a value that divides the given number exactly, with no remainder

  • e.g. 6 is a factor of 18

• A prime number is a number which has exactly two factors; itself and 1

  • e.g. 5 is a prime number, as its only factors are 5 and 1

  • You should remember the first few prime numbers:

    • 2, 3, 5, 7, 11, 13, 17, 19, …

• The prime factors of a number are therefore all the primes which multiply to give that number

  • e.g. The prime factors of 30 are 2, 3, and 5

    • 2 × 3 × 5 = 30

How do I find prime factors?

• Use a factor tree to find prime factors

  • Split the number up into a pair of factors

  • Then split each of those factors up into another pair

  • Continue splitting up factors along each "branch" until you get to a prime number

    • These can not be split into anything other than 1 and themselves

    • It helps to circle the prime numbers at the end of the branches

• One way to split each number up is by dividing it by the smallest possible prime number

  • This gives a prime factor at each stage

• Another way to split each number up by dividing by easy numbers such as 10

  • These do not need to be prime numbers

  • You will need to split any non-prime numbers into pairs of factors

  • These trees can get messy

• The number will always end up with the same numbers circled no matter which pairs of numbers you choose

• A number can be uniquely written as a product of prime factors

  • Write the prime factors as a multiplication, in ascending order

    • 360 = 2 × 2 × 2 × 3 × 3 × 5

  • This can then be written more concisely using powers

    • 360 = 2³ × 3² × 5

• A question asking you to do this will usually be phrased as "Express … as the product of its prime factors"