The factor calculator finds all the factors of a number. It list all the factor pairs and label the prime factors.

The calculator also tells you how many factors a number has and how many of those factors are prime (so you don't have to count them).

Here's a quick overview of what we mean by factors.

## What Are Factors?

A factor of a number is an integer that divides into that number evenly (leaves no remainder).

For example, $\hspace{0.2em} 2 \hspace{0.2em}$ is a factor of $\hspace{0.2em} 12 \hspace{0.2em}$. Similarly, $\hspace{0.2em} 3 \hspace{0.2em}$ is also a factor of $\hspace{0.2em} 8 \hspace{0.2em}$. But $\hspace{0.2em} 5 \hspace{0.2em}$ is not a factor of $\hspace{0.2em} 12 \hspace{0.2em}$ because it leaves a remainder, $\hspace{0.2em} 2 \hspace{0.2em}$.

Another way to think about factors is as follows.

If two or more integers multiply together to give a product, each of those integers is a factor of the product.

For example —

- $\hspace{0.2em} 5 \hspace{0.2em}$ x $\hspace{0.2em} 7 \hspace{0.2em}$ = $\hspace{0.2em} 35 \hspace{0.2em}$. So, both $\hspace{0.2em} 5 \hspace{0.2em}$ and $\hspace{0.2em} 7 \hspace{0.2em}$ are factors of $\hspace{0.2em} 35 \hspace{0.2em}$.
- $\hspace{0.2em} 9 \hspace{0.2em}$ x $\hspace{0.2em} 2 \hspace{0.2em}$ = $\hspace{0.2em} 18 \hspace{0.2em}$. So, both $\hspace{0.2em} 9 \hspace{0.2em}$ and $\hspace{0.2em} 2 \hspace{0.2em}$ are factors of $\hspace{0.2em} 18 \hspace{0.2em}$.

Example

Are $\hspace{0.2em} 6 \hspace{0.2em}$ and/or $\hspace{0.2em} 4 \hspace{0.2em}$ factors of $\hspace{0.2em} 18 \hspace{0.2em}$?

Solution

If we divide $\hspace{0.2em} 18 \hspace{0.2em}$ by $\hspace{0.2em} 6 \hspace{0.2em}$, there are no remainders. So $\hspace{0.2em} 6 \hspace{0.2em}$ is a factor of $\hspace{0.2em} 18 \hspace{0.2em}$.

On the other hand, $\hspace{0.2em} 4 \hspace{0.2em}$ does not divide into $\hspace{0.2em} 18 \hspace{0.2em}$ evenly. There is a remainder of $is not
2
)}. That means $\hspace{0.2em} 4 \hspace{0.2em}$ is not a factor of $\hspace{0.2em} 18 \hspace{0.2em}$.

Example

Find all the factors of $\hspace{0.2em} 72 \hspace{0.2em}$.

Solution

To find all the factors of a number, we split the number into all possible factors. Here's how.

Step 1. The first pair would be $\hspace{0.2em} 1 \hspace{0.2em}$ times the number itself.

$1 \hspace{0.25em} \times \hspace{0.25em} 72$

Step 2. For the next pair, we increase the first factor from $\hspace{0.2em} 1 \hspace{0.2em}$ to the next smallest factor, in this case, $\hspace{0.2em} 2 \hspace{0.2em}$.

$\begin{align*} 1 \hspace{0.25em} \times \hspace{0.25em} 72 \\[1em] 2 \hspace{0.25em} \times \hspace{0.25em} 36 \end{align*}$

Step 3. We keep moving ahead with progressively higher factors as long as the seond factor is smaller than or equal to the second factor.

Here's what the list would look like once we have completed the process.

$\begin{align*} 1 \hspace{0.25em} &\times \hspace{0.25em} 72 \\[1em] 2 \hspace{0.25em} &\times \hspace{0.25em} 36 \\[1em] 3 \hspace{0.25em} &\times \hspace{0.25em} 24 \\[1em] 4 \hspace{0.25em} &\times \hspace{0.25em} 18 \\[1em] 6 \hspace{0.25em} &\times \hspace{0.25em} 12 \\[1em] {\color{Red} 8} \hspace{0.25em} &\times \hspace{0.25em} {\color{Teal} 9} \\[1em] {\color{Teal} 9} \hspace{0.25em} &\times \hspace{0.25em} {\color{Red} 8} \end{align*}$

As you can see, the pairs start repeating as the first factor surpasses the second. So, going ahead would not give us any new factors.

Step 4. The table (with pairs) above has all the factors we were looking for. So, the factors of $\hspace{0.2em} 72 \hspace{0.2em}$ are —

$1, \hspace{0.4em} 2, \hspace{0.4em} 3, \hspace{0.4em} 4, \hspace{0.4em} 6, \hspace{0.4em} 8, \hspace{0.4em} 9, \hspace{0.4em} \text{and} \hspace{0.4em} 12$

For the negative factors, we place a negative sign before each of the factors above.

Now before we conclude, let me answer a question you might have.

Can Factors Be Negative?

Yes. Technically, factors can also be negative. So factors of 4 would include $\hspace{0.2em} -1 \hspace{0.2em}$, $\hspace{0.2em} -2 \hspace{0.2em}$, and $\hspace{0.2em} -4 \hspace{0.2em}$ along with $\hspace{0.2em} 1 \hspace{0.2em}$, $\hspace{0.2em} 2 \hspace{0.2em}$, and $\hspace{0.2em} 4 \hspace{0.2em}$. However, generally, when we say factors, we mean positive factors.