Prime Factorization
Every composite number is a unique product of primes.
About this topic
Prime factorization writes a composite number as a product of primes. A factor tree makes this visual: split the number into any two factors, then keep splitting until every branch ends in a prime. For 36 you get 2 × 2 × 3 × 3, written 2² × 3².
The Fundamental Theorem of Arithmetic says that the result is unique apart from ordering, so each number has exactly one prime fingerprint. You can start the tree with any factor pair, such as 4 × 9 or 6 × 6, and still end with the same primes. Another method is repeated division: divide by 2 as long as you can, then 3, then 5, and so on. For 360 this gives 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5.
This is the most reliable way to find greatest common divisors and least common multiples, and to simplify roots. For example, the square root of 36 is easy to see from 2² × 3², since each pair of equal primes comes out as a single factor: 2 × 3 = 6. Counting factors also becomes simple: add one to each exponent and multiply.
Students often stop too early, leaving a composite factor such as 4 on a branch. Every end of the tree must be prime. Check your work by multiplying the primes back together and confirming you reach the original number. Prime factorizations also appear in cryptography and in simplifying algebraic fractions.
Key ideas
- Use a factor tree to split numbers
- The Fundamental Theorem of Arithmetic guarantees uniqueness
- Write repeated primes with exponents
Quick check
Use cases: where Prime Factorization helps
- Homework: review the key idea behind Prime Factorization before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Prime Factorization without notes.
- Visual learning: play with the interactive lab to see how Prime Factorization changes when you adjust its inputs.
- Teaching a friend: explain Prime Factorization in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Prime Factorization?
Every composite number is a unique product of primes.
What will I learn in Prime Factorization?
Use a factor tree to split numbers. The Fundamental Theorem of Arithmetic guarantees uniqueness. Write repeated primes with exponents.
Is Prime Factorization part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Prime Factorization lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Prime Factorization?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.