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Prime Factorization Calculator
Break a whole number into its prime factors and see the result in exponent form (360 = 2³ × 3² × 5), with the division ladder, the number and sum of divisors and a list of every divisor. Works for numbers up to 10¹⁵.
Any whole number from 2 up to 10¹⁵ (1,000,000,000,000,000)
Enter your values to see the result.
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How to use the Prime Factorization Calculator
- Enter a whole number from 2 up to 10¹⁵, or tap an example.
- Press Find prime factors (results also update as you type).
- Read the exponent form and the expanded product of primes.
- Follow the factor ladder to see each division by a prime.
- Copy the results or download the prime factor table as CSV.
How prime factorization works
Divide by the smallest prime that divides the number evenly, write it down, and continue with the quotient. Stop when the quotient is 1. For example: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1, so 360 = 2³ × 3² × 5.
From the factorization n = p₁a × p₂b × … you can read off:
- Number of divisors: (a + 1)(b + 1)…
- Sum of divisors: (1 + p₁ + … + p₁a)(1 + p₂ + … + p₂b)…
- Euler's totient: φ(n) = n × (1 − 1/p₁)(1 − 1/p₂)…
- GCF of two numbers: the shared primes with the smaller exponents. LCM: every prime with the larger exponent.
Prime factorizations of common numbers
| Number | Prime factorization | Divisors |
|---|---|---|
| 12 | 2² × 3 | 6 |
| 24 | 2³ × 3 | 8 |
| 36 | 2² × 3² | 9 |
| 48 | 2⁴ × 3 | 10 |
| 60 | 2² × 3 × 5 | 12 |
| 72 | 2³ × 3² | 12 |
| 84 | 2² × 3 × 7 | 12 |
| 90 | 2 × 3² × 5 | 12 |
| 100 | 2² × 5² | 9 |
| 120 | 2³ × 3 × 5 | 16 |
| 144 | 2⁴ × 3² | 15 |
| 360 | 2³ × 3² × 5 | 24 |
| 1,000 | 2³ × 5³ | 16 |
| 1,001 | 7 × 11 × 13 | 8 |
The primes below 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.
Frequently asked questions
What is prime factorization?
Writing a whole number as a product of prime numbers. Every whole number greater than 1 has exactly one prime factorization (the fundamental theorem of arithmetic), for example 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
How do you find the prime factors of a number?
Divide by the smallest prime that goes in evenly (2, then 3, 5, 7 …) and repeat with the quotient until you reach 1. You only need to test primes up to the square root of what is left; if none divides it, the remainder is prime.
How do you count the divisors from the prime factorization?
Add 1 to each exponent and multiply. 360 = 2³ × 3² × 5¹ has (3+1)(2+1)(1+1) = 24 divisors.
What is a factor tree?
A diagram that splits a number into two factors, then splits each factor again until only primes remain. The ladder shown here is the same idea written as repeated division by primes.
Is 1 a prime number?
No. A prime has exactly two divisors, 1 and itself, and 1 has only one. Leaving 1 out keeps the prime factorization of every number unique.
What is the prime factorization of 100?
100 = 2² × 5² (2 × 2 × 5 × 5). It has (2+1)(2+1) = 9 divisors: 1, 2, 4, 5, 10, 20, 25, 50 and 100.
How is prime factorization used?
To find the greatest common factor and least common multiple, simplify fractions and square roots, count divisors, and in cryptography, where RSA relies on how hard it is to factor very large numbers.
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