Prime Factorization
Learn how to break down any number into its prime factor building blocks.
Definition
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Worked Examples
Find the prime factorization of 36.
Start with 36 and find any factor pair
(or , , etc.) → Choose
Factor each 6
(both are prime) → Circle 2 and 3
Collect all prime factors
→ Four prime factors
Write in exponential form
Two 2s and two 3s: →
Answer:
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 x 6, students might write 2 x 3 instead of 2 x 2 x 3 x 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 x 18 vs 6 x 6).
Correct: The final prime factorization is always the same, no matter which factor pair you start with!
Interactive Visual
Factor Tree
Enter a number to see its factor tree and prime factorization.
Venn Diagram - GCF & LCM
Prime factors of 12:
Prime factors of 18:
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)
Explore the factors of two numbers.
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
16 problemsWhat is the prime factorization of 12?
Why It Matters
- Simplifying fractions: Find common factors to reduce to
- 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
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!
A simple security code uses the product of two primes: .
Find the two prime factors.
Step 1: Write the mathematical expression
Find p and q where :
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 () can be divided into groups of 2, 3, 4, or 6.
A composer wants to find all the ways to divide 24 beats into equal groups.
First, find the prime factorization of 24.
Step 1: Write the mathematical expression
Write 24 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 number has exactly one prime factorization
- 4Write your answer in exponential form:
- 5Prime factorization helps with GCF, LCM, simplifying fractions, and cryptography
Frequently Asked Questions
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:
Formula Card
Exponential Form
Group repeated primes with exponents
Divisibility by 2
If the number ends in 0, 2, 4, 6, or 8
Divisibility by 3
If the sum of digits is divisible by 3
Divisibility by 5
If the number ends in 0 or 5