Prime Factorization

Learn how to break down any number into its prime factor building blocks.

Intermediate25 minLesson
Printable worksheet

Definition

Prime factorization is the process of expressing a number as a product of its prime factors.
Every whole number greater than 1 can be written as a product of prime numbers in exactly one way (only the order of the factors can change). This is called the Fundamental Theorem of Arithmetic.
For example:
36=2×2×3×3=22×3236 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2
The prime factorization of 36 shows that it is built from exactly two 2s and two 3s.

Try it now

What is the prime factorization of 12?

Worked Examples

Find the prime factorization of 36.
1

Start with 36 and find any factor pair

36=6×636 = 6 \times 6 (or 4×94 \times 9, 2×182 \times 18, etc.) → Choose 6×66 \times 6
2

Factor each 6

6=2×36 = 2 \times 3 (both are prime) → Circle 2 and 3
3

Collect all prime factors

36=2×3×2×336 = 2 \times 3 \times 2 \times 3 → Four prime factors
4

Write in exponential form

Two 2s and two 3s: 22×322^2 \times 3^2 → 36=22×3236 = 2^2 \times 3^2

Common Mistakes

Stopping before all factors are prime

Why it's wrong: Students sometimes stop at composite numbers like 4 or 9, thinking they are done.

Correct: Keep factoring until every number at the bottom of your tree is prime (2, 3, 5, 7, 11...).

Forgetting to include repeated factors

Why it's wrong: When 36=6×636 = 6 \times 6, students might write 2×32 \times 3 instead of 2×2×3×32 \times 2 \times 3 \times 3.

Correct: Count ALL prime factors from EVERY branch of the tree.

Thinking 1 is a prime number

Why it's wrong: 1 divides every number, but it is not considered prime.

Correct: Prime numbers start at 2. The number 1 is neither prime nor composite.

Getting different answers with different factor trees

Why it's wrong: Students worry when they start with different factors (2×182 \times 18 vs 6×66 \times 6).

Correct: The final prime factorization is always the same, no matter which factor pair you start with!

Interactive Visual

Factor Tree

Number:
Composite
Prime

Enter a number to see its factor tree and prime factorization.

Venn Diagram - GCF & LCM

Number A:
Number B:
121822, 33Only in ACommonOnly in B

Prime factors of 12:

223

Prime factors of 18:

233

GCF

6

2 × 3 = 6

LCM

36

2 × 2 × 3 × 3 = 36

How it works:

  • • GCF = product of common prime factors (center of diagram)
  • • LCM = product of all prime factors (entire diagram)

Enter two numbers to see their prime factorization in a Venn diagram.

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the prime factorization of 12?

Why It Matters

Prime factorization is a fundamental skill that unlocks many areas of mathematics:
  • Simplifying fractions: Find common factors to reduce 2436\frac{24}{36} to 23\frac{2}{3}
  • Finding GCF and LCM: Essential for adding fractions with different denominators
  • Cryptography: Modern internet security relies on prime factorization being difficult for large numbers
  • Problem solving: Many competition math problems use prime factorization
Think of primes as the "atoms" of numbers - they cannot be broken down further!

Real World Applications

Internet Security (RSA Encryption)

Online banking and secure websites use encryption based on multiplying two very large prime numbers. Finding the prime factors of this product is extremely difficult, which keeps your data safe.

Example:

A 2048-bit encryption key uses primes with over 300 digits. Even the fastest computers would take millions of years to factor it!

1Try It Yourself

A simple security code uses the product of two primes: p×q=91p \times q = 91.

Find the two prime factors.

Step 1: Choose the right answer

Which product of two primes equals 91?

Music and Rhythm

Musicians use prime factorization to understand time signatures and rhythmic patterns. Finding common factors helps in creating polyrhythms.

Example:

A rhythm in 12 beats (22×32^2 \times 3) can be divided into groups of 2, 3, 4, or 6.

2Try It Yourself

A composer wants to find all the ways to divide 24 beats into equal groups.

First, find the prime factorization of 24.

Step 1: Choose the right answer

Which is 24 written as a product of primes?

Key Takeaways

  • 1Prime factorization expresses a number as a product of prime numbers
  • 2Use a factor tree: keep splitting numbers until all branches end in primes
  • 3The Fundamental Theorem of Arithmetic: every whole number greater than 1 has exactly one prime factorization (apart from the order)
  • 4Write your answer in exponential form: 36=22×3236 = 2^2 \times 3^2
  • 5Prime factorization helps with GCF, LCM, simplifying fractions, and cryptography

Frequently Asked Questions

No! You can start with any factor pair and you will always get the same prime factorization. For example, 36=2×1836 = 2 \times 18 or 36=4×936 = 4 \times 9 or 36=6×636 = 6 \times 6 all lead to 22×322^2 \times 3^2.
No! You can start with any factor pair and you will always get the same prime factorization. For example, 36=2×1836 = 2 \times 18 or 36=4×936 = 4 \times 9 or 36=6×636 = 6 \times 6 all lead to 22×322^2 \times 3^2.
Then you're done! The prime factorization of a prime number (like 17) is just itself: 17=1717 = 17.
If 1 were prime, then every number would have infinitely many prime factorizations (36=2×2×3×3=1×2×2×3×3=1×1×2×…36 = 2 \times 2 \times 3 \times 3 = 1 \times 2 \times 2 \times 3 \times 3 = 1 \times 1 \times 2 \times \dots). To keep the Fundamental Theorem true, we define primes as starting at 2.

Glossary

Prime number
A number greater than 1 that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11)
Composite number
A number greater than 1 that has more than two factors (e.g., 4, 6, 8, 9, 10)
Factor tree
A diagram that breaks down a number into its prime factors by repeatedly splitting it
Prime factorization
Writing a number as a product of prime numbers
Exponential form
Using exponents to show repeated factors: 2×2×2=232 \times 2 \times 2 = 2^3

Formula Card

Exponential Form

2×2×3=22×32 \times 2 \times 3 = 2^2 \times 3

Group repeated primes with exponents

Divisibility by 2

Even⇒÷2\text{Even} \Rightarrow \div 2

If the number ends in 0, 2, 4, 6, or 8

Divisibility by 3

Digit sum divisible by 3⇒÷3\text{Digit sum divisible by } 3 \Rightarrow \div 3

If the sum of digits is divisible by 3

Divisibility by 5

Ends in 0 or 5⇒÷5\text{Ends in 0 or 5} \Rightarrow \div 5

If the number ends in 0 or 5

Today's challengeOne problem, three tries

More in This Topic

Factors and Multiples

Elementary20 min

Prime and Composite Numbers

Elementary20 min

Greatest Common Factor (GCF)

Intermediate25 min

Least Common Multiple (LCM)

Intermediate25 min