# Prime Factorization Calculator

Find the prime factors for a number, including large numbers, by entering the number below. The calculator will also indicate if the number is a prime number or not.

## Prime Factors:

2, 2, 3, 11

### Solution

132 = 2 × 2 × 3 × 11
132 = 22 × 31 × 111
Learn how we calculated this below

## How To Find Prime Factors

The prime factors of a number are a set of prime numbers that, when multiplied together, equal the number.

A prime number is a number that is only divisible by itself and 1.

Prime factorization of a number is the act of finding all of the prime factors of a number. This process is also known as Prime Decomposition.

There is no mathematical formula for finding all the prime factors of any number. In fact, mathematicians wonder if there is a largest prime number and are using computer algorithms to try and find it if it exists, but so far, one has not been found.

### Trial Division Method

One method of prime factorization is called the trial division method. To use this method to find all the prime factors of a number, say x, for example, start by dividing x by any prime number.

If the resulting value is not a whole number, then that prime number is not a prime factor of x. If it is, then divide the result by another prime number. Continue this process until the final result is also a prime number.

That is, the final result should only be divisible by itself and 1

This method is often visualized using a factor tree (see the diagram below for an example). The original number sits atop the tree, and its divisors are the branches. The ends of all the branches are the prime factors of the original number.

How do we know which prime numbers to use when dividing the original number? This will take some trial and error, hence the name Trial Division Method. A good strategy is to determine if the number is even or odd.

If even, start by dividing the number by 2 and continue to do so until the resulting value is odd, then try to divide by 3, then 5, and so on. Any prime number that you use to divide either the original number or a resulting value is a prime factor of the original number.

To know which numbers are prime that you should try dividing the original number by, see the list of all of the prime numbers under 1000 below as a helpful reference.

As you divide the original number and resulting factors by prime numbers following the example below, you’ll see that you will be creating your own factor tree!

You might also find our greatest common factor calculator helpful to find all factors for a number.

For example, let’s find the prime factors of the number 132.

Since 132 is an even number, start by dividing 132 by 2, which equals 66. 2 is a prime number and cannot be divided further, so it is a prime factor.

However, 66 can be divided by 2 again to get 33. Now we have another 2 as a prime factor.

We can divide 33 by 3 to get 11. 3 is a prime number, so it’s also a prime factor. 11 cannot be divided again and so it is also a prime factor.

Putting it all together, the prime factors of 132 are [11,3,2,2]. These are all prime numbers, and if we multiply them all together, we’ll get 132.

## List of Prime Factors Under 1,000

 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997

### Do all numbers have prime factors?

Yes! In fact, the Fundamental Theorem of Arithmetic states that every integer greater than one can be written as a product of prime numbers.

### What is prime factorization used for?

Prime factorization is used for finding the least common multiple of multiple numbers, for example, when combining fractions. In addition, it’s also used in cryptography to help protect digital codes on computers.

### Can negative numbers be prime factors?

By definition, all prime numbers are positive, so generally no, negative numbers cannot be prime factors. However, prime numbers can be negative if they are defined such that they are prime if they can only be divided by -1 and itself.

Except for rare cases, you probably won’t encounter negative prime factors.