Prime Factor Decomposition (Cambridge International IGCSE Mathematics)

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

Exam Hint Exam questions on this topic will ask you to express a number as a product of its prime factors. You should simplify your final answer by using indices. You need to show your working clearly to gain full marks.

Worked Example Express 432 as a product of powers of its prime factors. Show your working clearly.

Answer:

Create a factor tree Start with 432 and choose any two numbers that multiply together to make 432

• 2 and 216 is an obvious pair

Continue to split non-prime numbers into pairs of factors

The answer will be the same regardless of the factors chosen in the first step

Write the prime numbers out as a product

$432=2\times2\times2\times2\times3\times3\times3$

Any repeated prime factors can be written as a power

Final answer: $432=2^{4}\times3^{3}$