Prime Factor Decomposition

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What are prime factors?

Factors are things that are multiplied together
Prime numbers are numbers which have exactly two factors
- Themselves and 1
The prime factors of a number are therefore all the prime numbers which multiply to give that number
You should remember the first few prime numbers:
- 2, 3, 5, 7, 11, 13, 17, 19, …

How do I find the prime factors?

Use a FACTOR TREE to find prime factors
- Split a number up into a pair of factors which multiply to give the number
- Continue splitting up numbers until you get to a prime number
  - These can not be split into anything other than 1 and themselves
A number can be uniquely written as a product of prime factors
- Write the prime factors IN ASCENDING ORDER with × between
  - e.g. 72 = 2 × 2 × 2 × 3 × 3
- Write with POWERS if asked
  - e.g. 72 = 2³ × 3²

How might a question be worded?

This is one of those topics where questions can use different phrases that all mean the same thing …

Express … as the product of prime factors
Find the prime factor decomposition of …
Find the prime factorisation of …

Worked example

Find the prime factors of 360.

Give your answer in the form $2^{p} \times 3^{q} \times 5^{r}$ , where $p$ , $q$ and $r$ are integers to be found.

For each number find any two numbers, (not 1), which are factors and write those as the next pair of numbers in the tree.

If a number is prime, put a circle around it.

When all the end numbers are circled, you are done!

Factor-Tree-360, IGCSE & GCSE Maths revision notes

Write down all of the circled numbers, don't miss any of the repeated ones.

For any numbers that are repeated, write them as powers of the number.

$360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$

You don't usually have to write a "1" as a power if there is a number that isn't repeated, but in this question, it has asked for it.

$360 = 2^{3} \times 3^{2} \times 5^{1}$

Uses of Prime Factor Decomposition

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 $2 \times 3 \times \sqrt{2 \times 5} = 6 \sqrt{10}$

Worked example

$N = 2^{3} \times 3^{2} \times 5^{7}$ and $A N = B$ where $A$ is an integer and $B$ is a non-zero square number.

Find the smallest value of $A$ .

Substitute N = 2³× 3²× 5⁷ into the formula AN = B.

A(2³× 3²× 5⁷ ) = B

2, 3 and 5 are all prime numbers, so for A(2³× 3²× 5⁷ ) to be a square number, its prime factors must all have even powers.

Consider the prime factors A needs to have to make all the values on the left hand side have even powers.

(2 × 5) (2³× 3²× 5⁷) = B

2⁴× 3²× 5⁸= B

So A when written as a product of its prime factors, is 2 × 5.

Make sure you A as an integer value in the answer.

A = 10

Prime Factor Decomposition (Cambridge O Level Maths)

Revision Note