Prime Factor Decomposition (Edexcel GCSE Maths: Foundation)

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

• 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"

When a number has been written as in its prime factor decomposition (PFD), it can be used to find out if that number is a square or cube number, or to find the square root of that number without using a calculator.

How can I use PFD to tell if a number is a square or a cube number?

• If all the indices in the prime factor decomposition of a number are even, then that number is a square number
• For example, the prime factor decomposition of 7056 is 2⁴ × 3² × 7², so it must be a square number
• If all the indices in the prime factor decomposition of a number are multiples of 3, then that number is a cube number
• For example, the prime factor decomposition of 1728000 is 2⁹ × 3³ × 5³, so it must be a cube number

How can I use PFD to find the square root of a square number?

• Write the number in its prime factor decomposition
  • All the indices should be even if it is a square number
• Halve all of the indices
• This is the prime factor decomposition of the square root of the number
  • If you need to write the square root as an integer then multiply the prime factors together

How can I use PFD to find the exact square root of a number?

• Steps to find the square root of any other number which has been written as a product of its prime factors
  • STEP 1
    Write the prime factors out as individual factors
  • STEP 2
    Pair the factors together so that any two prime factors that are the same can be written just once as a power of 2
  • STEP 3
    Find the product of each of these paired prime factors, ignoring that each one is squared
    • This number will be written as an integer in front of the square root sign
  • STEP 4
    Multiply the remaining factors together
    • None of your remaining factors should be the same
    • This number will go inside the square root symbol
  • STEP 5
    Write the answer as a product of the integer from step 3 and the square root of the integer from step 4
  • For example, the prime factor decomposition of 360 is 2³ × 3² × 5
    • This can be written as 2² × 2 × 3² × 5 or 2²  × 3² × 2 × 5
    • So the exact square root of 360 is