# Factoring Calculator

## Using the Factoring Calculator

This factoring calculator takes as input a positive integer and uses trial division to determine all of the factors of that number. To find the factors for a number, simply enter it at the top of the calculator and it will be decomposed instantaneously.

The center portion of the factoring calculator has each of the factors shown in a colored box. Every factor has a companion number that when multiplied factor together, gives the original input as a product. Each such set of two numbers is called a factor pair. Drag the mouse pointer over one component of the pair in the factoring calculator, the other component of the factor pair will be highlighted, and the corresponding multiplication expression will appear. It’s a lot of fun to explore numbers and see how different factor pairs create their products.

A prime number will have only one factor pair consisting of the number one and the prime number itself. Composite numbers have multiple factor pairs.

Unlike prime factorization, regular number factoring produces all of the potential factors of a number, not just the prime factors. That means the list of values returned by this factoring calculator can be surprisingly long for some numbers that are highly composite.

The factoring calculator will highlight the first few pairs of factor pairs using a unique color. If you use the factoring calculator to factor a number that is very composite (it has a large number of factors), the factors after the first several will be colored gray. However, hovering over these factors in the interactive part of factoring calculator will still show you the other paired factor and the multiplication fact.

If you are using this factoring calculator on a projector or Smart Board in a classroom setting, try clicking the ‘Zoom’ button and it will make the calculator display better formatted for presentations.

## How to Factor a Number

This factoring calculator uses a technique called “trial division” to find the factors for a number. This works by testing all of the divisors between one and the square root of the number. If a number divides evenly into the target number, it is a factor. If it does not divide evenly, the number is not a factor.

As an example, let’s calculate the factors of 48…

## Factors of 48

We know that 1 and 48 are factors of 48, so our list starts with those. We also know the square root of 48 is a number between 6 and 7, because 6^{2} is 36 and 7^{2} is 49 and 48 is in between those values. So we need to test the divisors from 2 through 7 to see if they are divisible evenly into 48….

48 ÷ 3 = 16

48 ÷ 4 = 12

48 ÷ 5 = 9 r 3

48 ÷ 6 = 8

48 ÷ 7 = 6 r 6 Evenly divisible, so 2 and 24 are factors of 48

Evenly divisible, so 3 and 16 are factors of 48

Evenly divisible, so 4 and 12 are factors of 48

NOT evenly divisible, so 5 is not a factor of 48

Evenly divisible, so 6 and 8 are factors of 48

NOT evenly divisible, so 6 is not a factor of 48

We can combine all of these numbers and get a complete list…

The factors of 48 are 1, 2, 3, 4, 6, 8. 1, 16, 24, 48

This is exactly the process the factoring calculator uses to determine the factors for a given number. While you can see the steps individually aren’t too bad, when dealing with larger numbers the division can become quite tedious and using something like this factoring calculator to factor larger integers is a handy tool when factoring for applications.

## When do I need to Factor Numbers?

There are many problems both in mathematics and in real-world application where factoring comes into play, and using a factoring calculator will often be a time saver when dealing with non-trivial values.

Factoring polynomial expressions is a common activity in algebra, and typically classroom problems will deal with coefficients that are easily factored. However, many students don’t realize these same procedures in engineering or scientific applications may have coefficients that are too large to be practically factored by hand. In the past, printed factor tables may have been used to solve these problems, but today a computer can factor very large numbers using trial division very quickly. This factoring calculator runs entirely in the web browser and factors large numbers in real time!

Factoring also makes up an important part of most modern cryptographic algorithms used to secure communications over the internet. While you can see a small number can be easily factored, very large numbers may require a significantly larger amount of computer time for a factoring algorithm to run, even one that is optimized for larger integers. To get a sense of this, try putting a highly composite number (720720 is a good example) into the factoring calculator and observe how many factor pairs are generated. Increase that value by a multiple of ten (so 7207200) and you can see in the results how much more work is involved! While this factoring calculator will only handle values up to nine digits (Sorry NSA!) you can imagine when factoring extremely large numbers, like those with thousands of digits, that the amount of work would be quite extensive… In fact, a factoring calculator that attempted to determine all of the factors of very long numbers would take many years to determine the factors for even one number, and probably wouldn’t have enough memory to even store the answer when it did!

## Comparing the Prime Factorization and Factoring Calculator Results

You can also plug an integer into the prime factorization calculator on this site and get more insights into its internal composition. While the prime factorization will not show all of the composites, but it does give insight into why how some composite integers relate to others to compose the integer being factored. Be sure to check out the Prime Factorization Calculator… It’s a lot of fun to experiment with as well!

## More Calculator and Worksheet Resources

When you are done playing with this factoring calculator, be sure to check out the other calculators as well as some of the factoring worksheets on the site. You’ll find links to some of these tools below, or up in the ‘Tools’ menu anywhere on the site!