Prime Factorization, LCM, GCF • Topic 1 of 3

Prime Factorization Using Factor Trees

What is Prime Factorization?

Prime factorization is breaking down a number into its prime factors – the prime numbers that multiply together to make the original number.

Prime Numbers: Numbers greater than 1 with exactly 2 factors (1 and itself)

  • Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Composite Numbers: Numbers with more than 2 factors

  • Examples: 4, 6, 8, 9, 10, 12...

What is a Factor Tree?

A factor tree is a diagram that shows how a number breaks down into prime factors.

Steps to Build a Factor Tree:

  1. Write the number at the top
  2. Find two factors that multiply to make the number
  3. Branch them down
  4. Continue breaking composite numbers until all "leaves" are prime
  5. Circle the prime numbers
FACTOR TREE FOR 24:

 24
 / 
 4 6
 / / 
 2 2 2 3
 
 Prime factors: 2, 2, 2, 3
 Written as: 2³ × 3


FACTOR TREE FOR 36:

 36
 / 
 6 6
 / / 
 2 3 2 3
 
 Prime factors: 2, 2, 3, 3
 Written as: 2² × 3²


FACTOR TREE FOR 48:

 48
 / 
 6 8
 / / 
 2 3 2 4
 / 
 2 2
 
 Prime factors: 2, 2, 2, 2, 3
 Written as: 2⁴ × 3


ANOTHER WAY TO BUILD (different factors):

 48
 / 
 4 12
 / / 
 2 2 3 4
 / 
 2 2
 
 Same result: 2⁴ × 3
Solved Examples
1
Worked Example

Find the prime factorization of 30 using a factor tree.

Solution

``` 30 / 5 6 / 2 3

Prime factors: 5, 2, 3 Answer: 2 × 3 × 5 ```

2
Worked Example

Find the prime factorization of 72.

Solution

``` 72 / 8 9 / / 2 4 3 3 / 2 2

Prime factors: 2, 2, 2, 3, 3 Answer: 2³ × 3² ```

3
Worked Example

Find the prime factorization of 100.

Solution

``` 100 / 10 10 / / 2 5 2 5

Prime factors: 2, 2, 5, 5 Answer: 2² × 5² ```

Key Points

  • Prime numbers have exactly 2 factors
  • Factor tree breaks numbers down to primes
  • Order doesn't matter – multiplication is commutative
  • Use exponents for repeated factors
  • Every number has exactly one prime factorization
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A prime number has exactly:
Explanation: $1$ and itself.
Q2.The prime factorization of $12$ is:
Explanation: $2^2\times3$.
Q3.The only even prime number is:
Explanation: $2$.
Q4.The smallest prime number is:
Explanation: $2$.
Practice more MCQs → 📝 Assignment · 35 marks