Math & Statistics
Prime Factorization Calculator – Find Prime Factors of Any Number
Use our free Prime Factorization Calculator to quickly find all prime factors of a number. Perfect for students, teachers, and anyone learning about prime numbers and factorization.
What is Prime Factorization?
Prime factorization is the process of breaking down a number into the product of its prime factors — numbers that can only be divided by 1 and themselves. Every positive integer greater than 1 has a unique prime factorization, a principle known as the Fundamental Theorem of Arithmetic.
For example, 360 = 2³ × 3² × 5. This is useful for simplifying fractions, finding GCD/LCM, solving number theory problems, and understanding the building blocks of numbers in cryptography and computer science.
How to Use This Calculator
- Enter any positive integer from 2 to 1,000,000
- Click Find Prime Factors
- View the prime factorization, division steps, and all divisors
- Use the exponential form for compact notation
Common Use Cases
- Simplifying fractions
- Finding GCD & LCM
- Number theory homework
- Cryptography basics
- Divisibility problems
- Competition math prep
First 25 Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
A prime number has exactly two factors: 1 and itself. The number 1 is not considered prime.
Common Prime Factorizations
| Number | Prime Factors | Exponential Form | Total Divisors |
|---|---|---|---|
| 12 | 2 × 2 × 3 | 2² × 3 | 6 |
| 36 | 2 × 2 × 3 × 3 | 2² × 3² | 9 |
| 60 | 2 × 2 × 3 × 5 | 2² × 3 × 5 | 12 |
| 100 | 2 × 2 × 5 × 5 | 2² × 5² | 9 |
| 360 | 2 × 2 × 2 × 3 × 3 × 5 | 2³ × 3² × 5 | 24 |
| 1000 | 2 × 2 × 2 × 5 × 5 × 5 | 2³ × 5³ | 16 |
How to Find Prime Factors (Division Method)
| Step | Action | Example (84) |
|---|---|---|
| 1 | Start with the smallest prime (2) | 84 ÷ 2 = 42 |
| 2 | Keep dividing by 2 while even | 42 ÷ 2 = 21 |
| 3 | Move to next prime (3) | 21 ÷ 3 = 7 |
| 4 | 7 is prime, stop | 7 ÷ 7 = 1 |
| Result | 84 = 2² × 3 × 7 | |
FAQ – Prime Factorization Calculator
What does the Prime Factorization Calculator do?
The Prime Factorization Calculator breaks down any positive integer into its prime factors — numbers that can only be divided by 1 and themselves. It shows the result as a product of prime numbers, e.g., 60 = 2 × 2 × 3 × 5.
What is prime factorization?
Prime factorization is the process of expressing a number as a multiplication of its prime numbers. For example, the prime factorization of 84 is 2 × 2 × 3 × 7. Every positive integer greater than 1 has a unique prime factorization.
Why is prime factorization important?
Prime factorization is useful in various fields of mathematics, including simplifying fractions, finding greatest common divisors (GCD), least common multiples (LCM), and cryptography (like RSA encryption algorithms).
What are prime numbers?
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, 17, etc. The number 2 is the only even prime number.
Can it show the exponential (power) form of factors?
Yes. In addition to listing repeated factors, the calculator displays prime factors in exponential form — for example, 360 = 2³ × 3² × 5. This compact notation is standard in mathematics.
What's the difference between factorization and prime factorization?
Factorization lists all possible factors (e.g., 12 has factors 1, 2, 3, 4, 6, 12), while prime factorization lists only the prime numbers whose product equals the number (12 = 2 × 2 × 3).
How are the total number of divisors calculated?
From the prime factorization, add 1 to each exponent and multiply them together. For 360 = 2³ × 3² × 5¹, the divisor count is (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 divisors.
Is 1 a prime number?
No. By mathematical convention, 1 is not considered a prime number. A prime must have exactly two distinct positive divisors: 1 and itself. The number 1 has only one divisor (itself), so it doesn't qualify.
What is the Fundamental Theorem of Arithmetic?
It states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This means prime factorization is always unique for any given number.
Can I use this for negative numbers?
This calculator works with positive integers from 2 to 1,000,000. For negative numbers, the prime factorization of the absolute value applies, with a −1 factor prepended.