Enter any positive integer in this prime factorization calculator and instantly get its prime factor decomposition, with grouped exponents and step-by-step division. Click Factorize Number to see the result.
Prime Factorization Calculator
Prime factor decomposition
Enter a positive integer greater than 1.
How to use the calculator
- In the Which number do you want to factorize? field, enter a positive integer greater than or equal to 2.
- Click Factorize Number. The Factorization field will display the result with grouped exponents.
- The step-by-step appears just below, showing each division performed.
- For a new calculation, click Clear.
What is prime factorization
Factorizing a positive integer means rewriting it as a product of prime numbers. This representation is unique for each integer greater than 1 — that is what the Fundamental Theorem of Arithmetic guarantees.
For example, 360 can be written as 2³ × 3² × 5. No matter the order in which you divide, the set of prime factors and their exponents will always be the same.
Prime numbers such as 7, 13 or 97 are their own factorization, since they have no divisors other than 1 and themselves.
How to calculate factorization manually
The most common method in school is successive division by primes:
- Write the number to the left of a vertical bar.
- Divide by the smallest prime that divides it (start with 2); write the divisor on the right and the quotient below the original number.
- Repeat the process with the quotient until it equals 1.
- The divisors used are the prime factors; group equal ones using exponents.
Example: factorize 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- 72 = 2³ × 3²
Factorization examples
| Number | Factorization | Note |
|---|---|---|
| 12 | 2² × 3 | Three factors: two 2s and one 3 |
| 60 | 2² × 3 × 5 | Product of three distinct primes |
| 100 | 2² × 5² | Perfect square |
| 360 | 2³ × 3² × 5 | Basis for many GCD and LCM problems |
| 97 | 97 | Prime number — its own factorization |
Frequently asked questions about factorization
What is prime factorization?
Factorizing a positive integer means writing that number as a product of prime numbers. For example, 60 = 2² × 3 × 5. This process is unique for each integer greater than 1, according to the Fundamental Theorem of Arithmetic.
Does every number have a prime factorization?
Yes. The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 can be factorized into primes in a unique way (disregarding the order of factors). Prime numbers are their own factorization — for example, 7 = 7.
How is factorization used to calculate GCD and LCM?
The GCD of two numbers is the product of the common prime factors taken with the smallest exponent. The LCM uses all factors that appear in either number, with the largest exponent. For example, for 12 = 2² × 3 and 18 = 2 × 3²: GCD = 2 × 3 = 6 and LCM = 2² × 3² = 36.
Does the number 1 have a prime factorization?
The number 1 is a special case: by convention, it is not prime and its product of prime factors is considered the empty product, equal to 1. The Fundamental Theorem of Arithmetic applies only to integers greater than 1.
What is factorization used for in everyday life?
Factorization is fundamental for calculating GCD and LCM, simplifying fractions, solving equations and identifying whether a number is prime. In cryptography, the difficulty of factorizing very large numbers is the basis of algorithms such as RSA.
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