Start With a Composite Number
A composite number can be split into smaller factors because it has more than two positive factors.
Free Math Guide
Prime Factorization Explained
Learn how to identify prime and composite numbers, find factor pairs, build factor trees, list repeated prime factors, and write prime factorizations using exponent notation.
Prime factorization means writing a composite whole number as a product of prime numbers. The prime factors act like the number's basic building blocks. Different factor trees may look different, but a number always has the same prime factorization apart from the order of the factors.
On This Page
- What prime factorization means
- Prime numbers and composite numbers
- Finding factor pairs
- Building a factor tree
- Knowing when to stop factoring
- Listing the prime factors
- Writing prime factors with exponents
- Worked prime-factorization examples
- Common prime-factorization mistakes
- Why prime factorization matters
- Prime-factorization checklist
- Interactive practice
What Is Prime Factorization?
Prime factorization is the process of rewriting a composite number as a product of prime numbers.
36 = 2 × 2 × 3 × 3 = 2² × 3²
End With Prime Numbers
Continue factoring until every number in the product is prime.
Prime Numbers Are the Building Blocks
Just as a whole number can be built by multiplying factors, every composite number can be built from a specific collection of prime factors.
Example: Prime Factorization of 36
- Split 36 into any valid factor pair.
- One choice is 4 × 9.
- Factor 4 as 2 × 2.
- Factor 9 as 3 × 3.
- Every factor is now prime.
36 = 2 × 2 × 3 × 3 = 2² × 3²
Prime Numbers and Composite Numbers
Before building factor trees, it is important to know whether a number is prime or composite.
Prime Number
A prime number has exactly two positive factors: 1 and itself.
2
3
5
7
11
13
Composite Number
A composite number has more than two positive factors and can be split into smaller whole-number factors.
4
6
8
9
10
12
The Number 1 Is Special
The number 1 is neither prime nor composite because it has only one positive factor.
Is 17 Prime or Composite?
The only positive factors of 17 are 1 and 17.
17 is prime
Is 18 Prime or Composite?
The number 18 has several factor pairs, including 1 × 18, 2 × 9, and 3 × 6.
18 is composite
Finding Factor Pairs
A factor pair is a pair of whole numbers whose product equals the original number.
Factor pairs of 24
1 × 24
2 × 12
3 × 8
4 × 6
Any Nontrivial Factor Pair Can Start the Tree
For a factor tree, avoid using 1 and the original number. Choose any other valid factor pair and keep factoring from there.
First Possible Start
24 = 2 × 12
Continue by factoring 12.
Another Possible Start
24 = 4 × 6
Continue by factoring both 4 and 6.
Different Starts Give the Same Prime Factors
A factor tree may begin with 2 × 12, 3 × 8, or 4 × 6. After every composite number is fully factored, each tree will produce the same prime factorization.
How to Build a Factor Tree
A factor tree breaks a composite number into smaller factors until every branch ends in a prime number.
1
Choose a Factor Pair
Split the original number into any two whole-number factors greater than 1.
2
Check Each Factor
Decide whether each new factor is prime or composite.
3
Split Composite Factors
Any composite factor must be broken into another factor pair.
4
Stop at Prime Leaves
A branch is complete when the number at the end is prime.
-
60
-
6
- 2
- 3
-
10
- 2
- 5
-
6
Worked Example: Factor 60
- Start with 60.
- Split 60 into 6 × 10.
- Since 6 is composite, factor it as 2 × 3.
- Since 10 is composite, factor it as 2 × 5.
- The leaves 2, 3, 2, and 5 are all prime.
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
The Shape of the Tree Can Vary
You could also begin with 60 = 3 × 20, 4 × 15, 5 × 12, or another valid factor pair. The final prime factors will still be the same.
When Do You Stop Factoring?
Stop only when every leaf at the bottom of the tree is prime.
Incomplete Tree
24 = 4 × 6
Both 4 and 6 are composite, so the tree is not finished.
Complete Factorization
24 = 2 × 2 × 2 × 3
Every factor is prime, so the factorization is complete.
Do Not Stop at a Composite Number
Numbers such as 4, 6, 8, 9, 10, 12, and 15 are not valid final leaves because they can still be factored.
Quick Prime Check
Before stopping at a leaf, ask whether it has any positive factor other than 1 and itself.
7 is prime
11 is prime
9 = 3 × 3
14 = 2 × 7
Listing the Prime Factors
Once the tree is complete, collect every prime leaf. Repeated primes must be included each time they appear.
Example: Prime Leaves of 72
2
2
2
3
3
72 = 2 × 2 × 2 × 3 × 3
Repeated Factors Matter
If 2 appears three times in the tree, write 2 three times in the expanded prime-factor list.
Order Does Not Matter
The products 2 × 2 × 3 × 5 and 5 × 3 × 2 × 2 represent the same prime factorization.
Include Every Prime Leaf
Leaving out even one repeated prime changes the product and gives the wrong original number.
Check by Multiplying
2 × 2 × 2 × 3 × 3 = 72
Multiplying the prime factors should reproduce the original number.
Writing Prime Factors with Exponents
Instead of writing the same prime factor multiple times, mathematicians use exponents. The exponent tells you how many copies of the prime factor are being multiplied together.
2 × 2
→
2²
3 × 3 × 3
→
3³
2 × 2 × 2 × 5
→
2³ × 5
What Does the Exponent Mean?
The exponent tells you how many times the base appears as a factor. For example, 2³ means 2 × 2 × 2, not 2 × 3.
Example
72 = 2 × 2 × 2 × 3 × 3
72 = 2³ × 3²
Worked Prime Factorization Examples
These examples show the complete process from the original number to the finished prime factorization.
Example 1
18
- 18 = 2 × 9
- 9 = 3 × 3
18 = 2 × 3²
Example 2
42
- 42 = 6 × 7
- 6 = 2 × 3
42 = 2 × 3 × 7
Example 3
96
- 96 = 12 × 8
- 12 = 2 × 2 × 3
- 8 = 2 × 2 × 2
96 = 2⁵ × 3
Example 4
180
- 180 = 18 × 10
- 18 = 2 × 3 × 3
- 10 = 2 × 5
180 = 2² × 3² × 5
Example 5
420
- 420 = 42 × 10
- 42 = 2 × 3 × 7
- 10 = 2 × 5
420 = 2² × 3 × 5 × 7
Notice the Pattern
Every example finishes with only prime numbers. The order of the prime factors does not matter, but every repeated prime must be included.
Common Prime Factorization Mistakes
Most mistakes happen because students stop factoring too early or forget that every final factor must be prime.
Mistake 1: Stopping Too Early
18 = 2 × 9
18 = 2 × 3²
Since 9 is composite, it must still be factored.
Mistake 2: Forgetting Repeated Prime Factors
24 = 2 × 3
24 = 2³ × 3
Every occurrence of a prime factor must be included.
Mistake 3: Calling 1 a Prime Number
The number 1 is neither prime nor composite. Prime numbers have exactly two positive factors.
Mistake 4: Leaving Composite Numbers in the Answer
36 = 4 × 9
36 = 2² × 3²
Your final answer should contain only prime numbers.
Where Prime Factorization Is Used
Prime factorization is much more than an isolated skill. It is used throughout middle school, high school, and college mathematics.
Simplifying Fractions
Identify common prime factors in the numerator and denominator before reducing.
Greatest Common Factor
Compare prime factorizations to find the largest common product.
Least Common Multiple
Combine prime factors to build the smallest common multiple.
Algebra and Number Theory
Factoring expressions and understanding divisibility both rely on prime factorization concepts.
Prime Factorization Checklist
Before you finish, ask yourself these five questions.
✓ Every factor is prime
No composite numbers remain.
✓ Every branch is complete
Every composite number was factored.
✓ Repeated primes included
Every repeated prime appears the correct number of times.
✓ Exponents are correct
Repeated factors are written using exponent notation.
✓ Product matches
Multiplying every prime factor reproduces the original number.
Ready to Practice?
Build factor trees, identify prime factors, and write prime factorizations with immediate feedback using the interactive Prime Factorization Practice tool.
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