Break a number into prime factors with a factor tree.
Every whole number above 1 is built from primes, and a prime factorization calculator shows exactly which ones, for any number from 2 up to 10 trillion. It shows the answer in exponent form, such as 2³ × 3² × 5, draws the factor tree, and lists each division step. It also tells you how many divisors the number has and what they add up to, which is useful in number theory and puzzle questions.
How to Use the Prime Factorization Calculator
- Enter a whole number of 2 or more.
- Read the prime factorization in exponent form and as a full list of factors.
- Scroll down to see the factor tree and the step-by-step divisions.
If the number is prime, the calculator tells you, because its only prime factor is itself.
What Is Prime Factorization?
A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Prime factorization writes a number as a product of primes. Every whole number above 1 has exactly one prime factorization, apart from the order of the factors. This is called the Fundamental Theorem of Arithmetic.
How Prime Factorization Works
Repeated division
- Divide the number by the smallest prime that goes into it exactly.
- Divide the result by the smallest prime that goes into it.
- Repeat until you reach 1.
For 360: 360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 ÷ 5 = 1
So 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Factor tree
A factor tree splits the number into any two factors, then splits each branch again until every branch ends in a prime. 360 can split into 36 × 10, then 6 × 6 and 2 × 5, then 2 × 3 for each 6. Any route gives the same primes at the ends of the branches.
Number of Divisors and Their Sum
Once you have the exponent form, you can count all the divisors without listing them. Add 1 to each exponent and multiply:
360 = 2³ × 3² × 5¹, so the number of divisors = (3 + 1) × (2 + 1) × (1 + 1) = 24.
The sum of all those divisors is (1 + 2 + 4 + 8) × (1 + 3 + 9) × (1 + 5) = 15 × 13 × 6 = 1,170.
Prime Factorization Examples
- 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7, with 12 divisors.
- 1,001 = 7 × 11 × 13
- 1,024 = 2¹⁰
- 97 is prime.
Divisibility Shortcuts
- 2: the last digit is even.
- 3: the digits add up to a multiple of 3. For 360, 3 + 6 + 0 = 9.
- 5: the last digit is 0 or 5.
- 11: the alternating sum of the digits is a multiple of 11. For 1,001: 1 − 0 + 0 − 1 = 0.
What Prime Factors Are Used For
- Finding the GCF and LCM of two numbers, by comparing their prime factors.
- Simplifying fractions and square roots. The square root calculator uses prime factors to simplify radicals.
- Cryptography. Online security such as RSA relies on the fact that factoring very large numbers is extremely slow.
Primes, GCF and LCM
- For a quick yes or no on whether a number is prime, use the prime number checker.
- For the greatest common factor of two or more numbers, use the GCF calculator. For the least common multiple, use the LCM calculator.
Prime Factorization FAQs
Is 1 a prime number?
No. A prime must have exactly two different factors, and 1 has only one. That is why 1 never appears in a prime factorization.
What is the prime factorization of a prime number?
A prime number’s factorization is just the number itself. For example, 97 = 97.
Why is there a limit of 10 trillion?
Above that size, numbers can be slow to factor in a web browser. The limit keeps results instant on phones and older computers.
How we checked this page: each factorization was confirmed by multiplying the primes back together, and the divisor counts and sums match the calculator.