Prime Factorization Calculator
Enter any positive whole number to find its prime factors, exponential form, CSV list, and a visual factor tree.
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Common Prime Factorization Reference Table
Common prime factorization reference lists the prime factors and exponential form for frequently used numbers. The number 100 is highlighted as the worked example used throughout this page.
| Number | Prime Factors | Exponential Form |
|---|---|---|
| 10 | 2 x 5 | 2^1 x 5^1 |
| 20 | 2 x 2 x 5 | 2^2 x 5^1 |
| 36 | 2 x 2 x 3 x 3 | 2^2 x 3^2 |
| 48 | 2 x 2 x 2 x 2 x 3 | 2^4 x 3^1 |
| 50 | 2 x 5 x 5 | 2^1 x 5^2 |
| 76 | 2 x 2 x 19 | 2^2 x 19^1 |
| 100 | 2 x 2 x 5 x 5 | 2^2 x 5^2 |
What Is Prime Factorization
What is prime factorization comes down to breaking a number apart into the prime numbers that multiply together to form it. This process is also known as prime decomposition or integer factorization, and every whole number greater than 1 has exactly one prime factorization.
A prime number is a whole number greater than 1 with only two factors, itself and 1. Numbers like 2, 3, 5, 7, 11, and 13 are prime, while numbers such as 4, 6, 8, 9, and 10 are not prime because they can be divided evenly by numbers other than 1 and themselves.
Why Prime Factorization Is Useful
Why prime factorization is useful becomes clear in topics like simplifying fractions, finding the greatest common factor, and finding the least common multiple. Breaking numbers into their prime building blocks makes these comparisons much simpler than working with the original numbers directly.
How to Find Prime Factorization by Trial Division
How to find prime factorization by trial division involves dividing the number repeatedly by the smallest possible prime until the result is 1. This method works for any whole number and is the same method used by this calculator.
- Start with the smallest prime number, 2, and divide the number by it if it divides evenly.
- Continue dividing the result by 2 until it no longer divides evenly.
- Move to the next prime number, 3, and repeat the process.
- Continue with each successive prime number until the remaining quotient is 1.
- List every prime divisor used, including repeats, as the prime factorization.
Worked Example for the Number 100
Worked example for the number 100 shows trial division in action. Dividing 100 by 2 gives 50, dividing 50 by 2 gives 25, then 25 is not divisible by 2 or 3 but is divisible by 5 to give 5, and dividing 5 by 5 gives 1. The prime factorization of 100 is therefore 2 times 2 times 5 times 5, or 2 squared times 5 squared in exponential form.
Reading Exponential Form
Reading exponential form means understanding that a small number written above and to the right of a prime, called an exponent, shows how many times that prime appears in the factorization. For example, 2 squared times 5 squared means 2 appears twice and 5 appears twice, which multiplies out to 4 times 25, or 100.
Using a Prime Factor Tree
Using a prime factor tree gives a visual way to see how a number splits into smaller factors at each step. Starting at the top with the original number, each branch splits into two factors, and branching continues until every branch ends in a prime number.
- The top of the tree is the original number being factored
- Each split shows two factors that multiply to give the number above
- Branches stop splitting once they reach a prime number
- Multiplying all the prime numbers at the bottom of the tree gives back the original number
Frequently Asked Questions
Prime factorization is the process of breaking a number down into the set of prime numbers that multiply together to produce the original number. This process is also called prime decomposition or integer factorization.
A prime number is a whole number greater than 1 that has only two factors, 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and 13, while numbers like 4, 6, 8, and 9 are not prime because they have additional factors.
To find the prime factorization of a number, divide it repeatedly by the smallest prime number that divides it evenly, starting with 2, then 3, then 5, and continuing until the remaining quotient is 1. The list of primes used in this process, including repeats, is the prime factorization.
The prime factorization of 100 is 2 times 2 times 5 times 5, which can also be written in exponential form as 2 squared times 5 squared. This means multiplying 2 by itself twice and 5 by itself twice gives 100.
No, 1 is not a prime number because a prime number must have exactly two distinct factors, 1 and itself, while 1 only has one factor. The number 1 is therefore excluded from prime factorization results.
A prime factor tree is a diagram that shows how a number breaks down step by step into smaller factor pairs until every branch ends in a prime number. The prime numbers at the ends of the branches, multiplied together, give the original number.