Prime Factorization Calculator
- Start with the number: 108.
- Divide by 2 twice: 108 ÷ 2 = 54, 54 ÷ 2 = 27.
- Divide by 3 three times: 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1.
- Multiply the primes back together: 2 × 2 × 3 × 3 × 3 = 108.
- Group equal factors: 2² × 3³.
- Smallest prime first: divide by 2 as long as it fits, then 3, then 5, and so on, until nothing is left but 1.
- Exponent form: equal primes are grouped, so 2 × 2 × 3 becomes 2² × 3.
- Number of divisors: add 1 to each exponent and multiply the results.
- Multiplying the factors gives 108.
- Divisors: (2 + 1) × (3 + 1) = 12. Their sum is 280.
- Numbers up to 108 that share no factor with it: 36 (Euler’s totient).
Prime Factorization Calculator: Find the Prime Factors of Any Number
This free prime factorization calculator breaks a whole number into the prime numbers that multiply together to make it. It shows the factors in a list and in exponent form, draws a factor tree, tells you whether the number is prime, and counts its divisors. Every division is written out, so the page also works as a quick way to check homework.
Quick answer: the prime factorization of 108 is 2 × 2 × 3 × 3 × 3, which is written 2² × 3³. It has 12 divisors.
What Are Prime Factors?
A prime number is a whole number greater than 1 whose only divisors are 1 and itself. The first few are 2, 3, 5, 7, 11 and 13. Any other whole number is composite, which means it can be split into smaller numbers. Prime factors are the primes you get when you keep splitting until no more splitting is possible. Every number above 1 has exactly one set of prime factors, which is why they are sometimes called the building blocks of numbers.
How to Find the Prime Factors of a Number
Divide the number by 2 as many times as it goes in evenly. Then try 3, then 5, then 7, and carry on through the primes. Each time a prime divides exactly, write it down and continue with the answer. Stop when the result is 1. You only need to test primes up to the square root of what is left, because if none of them fit, the remaining number is itself prime. The calculator follows exactly these steps and lists them in the working box.
| Number | Prime factorization | Divisors |
|---|---|---|
| 12 | 2² × 3 | 6 |
| 60 | 2² × 3 × 5 | 12 |
| 84 | 2² × 3 × 7 | 12 |
| 100 | 2² × 5² | 9 |
| 108 | 2² × 3³ | 12 |
| 360 | 2³ × 3² × 5 | 24 |
| 1001 | 7 × 11 × 13 | 8 |
| 9973 | 9973 (prime) | 2 |
Factor Trees
A factor tree is a picture of the same process. Write the number at the top, split it into two factors, and keep splitting any factor that is not prime. The numbers at the ends of the branches are the prime factors. Different trees for the same number can look different, but they always end with the same primes. The graph above draws the tree for numbers up to 99,999, and for larger numbers it shows the prime powers as blocks instead.
Exponent Form and Divisors
When a prime repeats, it is written with an exponent, so 2 × 2 × 2 becomes 2³. This short form makes the divisor count easy: add 1 to each exponent and multiply the results. For 2² × 3³ that is (2 + 1) × (3 + 1) = 12 divisors. The same idea gives the sum of divisors and Euler's totient, which are both shown in the Check your answer box.
Uses of Prime Factorization
Prime factors are used to simplify fractions, find the greatest common factor and the least common multiple, and to tell whether a number is a perfect square or a perfect cube. They also appear in number theory and in the codes that protect online payments. To simplify a fraction after factoring, the fraction calculator is a handy next step, and the math solver, the scientific calculator and the percentage calculator cover other everyday number work.
How to Use This Prime Factorization Calculator
Enter a number
Type a whole number, drag the slider, or tap one of the examples.
Read the factors
See the prime factors in list form and in exponent form.
Check the working
Follow each division in the step-by-step box.
Use the extras
See the factor tree, the divisor count and whether the number is prime.