# Prime Factorization Calculator

## Calculator Use

Use this prime numbers calculator to find all prime factors of a given integer number up to 10 trillion. This calculator presents:

- Prime factors of a number
- Prime decomposition in exponential form
- CSV (comma separated values) list of prime factors
- Factorization in a prime factors tree

For the first 5000 prime numbers, this calculator indicates the index of the prime number. The
*n ^{th}* prime number is denoted as Prime[n], so Prime[1] = 2, Prime[2] = 3, Prime[3] = 5, and so on.

The limit on the input number to factor is less than 10,000,000,000,000 (less than 10 trillion or a maximum of 13 digits).

## What is Prime Factorization?

Prime factorization or integer factorization of a number is breaking a number down into the set of prime numbers which multiply together to result in the original number. This is also known as prime decomposition.

## How to Find Prime Factorization of a Number

We cover two methods of prime factorization: find primes by trial division, and use primes to create a prime factors tree.

### Prime Factorization by Trial Division

Say you want to find the prime factors of 100 using trial division. Start by testing each integer to see if and how often it divides 100 and the subsequent quotients evenly. The resulting set of factors will be prime since, for example, when 2 is exhausted all multiples of 2 are also exhausted.

### Find the prime factors of 100:

- 100 ÷ 2 = 50; save
**2** - 50 ÷ 2 = 25; save
**2** - 25 ÷ 2 = 12.5, not evenly so divide by next highest number, 3
- 25 ÷ 3 = 8.333, not evenly so divide by next highest number, 4
- But, 4 is a multiple of 2 so it has already been checked, so divide by next highest number, 5
- 25 ÷ 5 = 5; save
**5** - 5 ÷ 5 = 1; save
**5**

List the resulting prime factors as a sequence of multiples, 2 x 2 x 5 x 5 or as factors with exponents, 2^{2} x 5^{2}.

### Examples of Prime Decomposition: Factors and Exponents

- Prime factorization of 100 is 2 x 2 x 5 x 5 or 2
^{2}x 5^{2} - Prime factorization of 76 is 2 x 2 x 19 or 2
^{2}x 19^{1} - Prime factorization of 50 is 2 x 5 x 5 or 2 x 5
^{2} - Prime factorization of 48 is 2 x 2 x 2 x 2 x 3 or 2
^{4}x 3^{1} - Prime factorization of 36 is 2 x 2 x 3 x 3 or 2
^{2}x 3^{2} - Prime factorization of 20 is 2 x 2 x 5 or 2
^{2}x 5^{1} - Prime factorization of 10 is 2 x 5 or 2
^{1}x 5^{1}

### Prime Factors Tree

Using a prime factorization tree to see the work, prime decomposition of 100 = 2 x 2 x 5 x 5 looks like this:

**2**

**2**

**5**

**5**

## Related Calculators

For a list of the first 1000 prime numbers see our See 1000 Prime Numbers Table.

## References and Further Reading

Math is Fun: Prime Factorization.

Weisstein, Eric W. "Prime Number." From* MathWorld*--A Wolfram Web Resource.

Weisstein, Eric W. "Direct Search Factorization." From* MathWorld*--A Wolfram Web Resource.

Weisstein, Eric W. "Prime Factorization Algorithms." From* MathWorld*--A Wolfram Web Resource.

Caldwell, Chris K. The Prime Pages.