Factors and How to Do Prime Factorization

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 $\hspace{0.2em} 8 \hspace{0.2em}$ . Similarly, $\hspace{0.2em} 4 \hspace{0.2em}$ 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 $\hspace{0.2em} 8 \hspace{0.2em}$ because it leaves a remainder, $\hspace{0.2em} 3 \hspace{0.2em}$ .

Here’s another way (and a more correct way) to think about factors.

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

For example -

$\hspace{0.2em} 2 \hspace{0.2em}$ x $\hspace{0.2em} 5 \hspace{0.2em}$ = $\hspace{0.2em} 10 \hspace{0.2em}$ . So, both $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 5 \hspace{0.2em}$ are factors of $\hspace{0.2em} 10 \hspace{0.2em}$ .
$\hspace{0.2em} 3 \hspace{0.2em}$ x $\hspace{0.2em} 8 \hspace{0.2em}$ = $\hspace{0.2em} 24 \hspace{0.2em}$ . So, both $\hspace{0.2em} 3 \hspace{0.2em}$ and $\hspace{0.2em} 8 \hspace{0.2em}$ are factors of $\hspace{0.2em} 24 \hspace{0.2em}$ .

Now before we move ahead, here's a question students often ask.

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.

Example

List all the factors of $\hspace{0.2em} 48 \hspace{0.2em}$ .

Solution

To find all the factors of a number, split the number into all possible factor pairs.

Step 1. Start with the most obvious pair. $\hspace{0.2em} 1 \hspace{0.2em}$ times the number itself.

1 \times 48

Step 2. For the next pair, in place of $\hspace{0.2em} 1 \hspace{0.2em}$ put the next factor. In this case, it would be $\hspace{0.2em} 2 \hspace{0.2em}$ . So,

\begin{align*} 1 \times 48 \\[1em] 2 \times 24 \end{align*}

Step 3. Keep replacing the first factor with larger numbers as long as it remains smaller than or equal to the second factor.

Here's what your list of factor pairs would look like.

\begin{align*} 1 &\times 48 \\[1em] 2 &\times 24 \\[1em] 3 &\times 16 \\[1em] 4 &\times 12 \\[1em] {\color{Red} 6} &\times {\color{Teal} 8} \\[1em] {\color{Red} 8} &\times {\color{Teal} 6} \end{align*}

So, we stop when the two factors bump into each other - just before the first factor overtakes the second. After that the factor pairs repeat, so no point in going on.

Step 4. The numbers making the pairs above are the factors we were looking for. So, the (positive) factors of $\hspace{0.2em} 48 \hspace{0.2em}$ are -

$\hspace{0.2em} 1 \hspace{0.2em}$ , $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} 3 \hspace{0.2em}$ , $\hspace{0.2em} 4 \hspace{0.2em}$ , $\hspace{0.2em} 6 \hspace{0.2em}$ , $\hspace{0.2em} 8 \hspace{0.2em}$ , $\hspace{0.2em} 12 \hspace{0.2em}$ , $\hspace{0.2em} 16 \hspace{0.2em}$ , $\hspace{0.2em} 24 \hspace{0.2em}$ , and $\hspace{0.2em} 48 \hspace{0.2em}$ .

If you need the negative factors too, just add a negative sign before each of them.

Example

Find all the factors of $\hspace{0.2em} 12 \hspace{0.2em}$ .

Solution

Again, we start by listing the factor pairs for 12.

\begin{align*} 1 &\times 12 \\[1em] 2 &\times 6 \\[1em] {\color{Red} 3} &\times {\color{Teal} 4} \\[1em] {\color{Red} 4} &\times {\color{Teal} 3} \end{align*}

And that gives us all the factors of $\hspace{0.2em} 12 \hspace{0.2em}$ - $\hspace{0.2em} 1 \hspace{0.2em}$ , $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} 3 \hspace{0.2em}$ , $\hspace{0.2em} 4 \hspace{0.2em}$ , $\hspace{0.2em} 6 \hspace{0.2em}$ , and $\hspace{0.2em} 12 \hspace{0.2em}$ .

Things to Remember

$\hspace{0.2em} 1 \hspace{0.2em}$ is a factor (and the smallest positive factor) of every number.
Every number is a factor of itself.
The largest factor of a positive number is the number itself. For example, $\hspace{0.2em} 12 \hspace{0.2em}$ is the largest factor of $\hspace{0.2em} 12 \hspace{0.2em}$ .
Factors of a factor are also factors. For example, $\hspace{0.2em} 12 \hspace{0.2em}$ is a factor of $\hspace{0.2em} 24 \hspace{0.2em}$ , and $\hspace{0.2em} 3 \hspace{0.2em}$ and $\hspace{0.2em} 4 \hspace{0.2em}$ are factors of $\hspace{0.2em} 12 \hspace{0.2em}$ . So $\hspace{0.2em} 3 \hspace{0.2em}$ and $\hspace{0.2em} 4 \hspace{0.2em}$ must also be factors of $\hspace{0.2em} 24 \hspace{0.2em}$ .

Prime Factorization

Now let’s look at a very important concept – prime factorization. It has many uses. For example, in finding the least common multiple (LCM) or greatest common factor (GCF) or simplifying fractions. So make sure you understand it well.

Let’s begin with a quick recap of what prime numbers are.

A prime number is any natural number that is greater than $\hspace{0.2em} 1 \hspace{0.2em}$ and has only two positive factors – $\hspace{0.2em} 1 \hspace{0.2em}$ and itself. For example, $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} 3 \hspace{0.2em}$ , $\hspace{0.2em} 5 \hspace{0.2em}$ , $\hspace{0.2em} 7 \hspace{0.2em}$ , $\hspace{0.2em} 11 \hspace{0.2em}$ , $\hspace{0.2em} 13 \hspace{0.2em}$ , $\hspace{0.2em} 17 \hspace{0.2em}$ , etc.

So prime numbers cannot be split into smaller factors. That's what makes them so important. They are the building blocks or all counting numbers greater than $\hspace{0.2em} 1 \hspace{0.2em}$ .

What is prime factorization?

By prime factorization, we mean splitting a number and expressing it as a product of its prime factors (factors that are prime numbers) only.

For example -

\begin{align*} 24 &= 2 \times 2 \times 2 \times 3 \\[1em] 90 &= 2 \times 3 \times 3 \times 5 \end{align*}

See how on the right side we have only prime factors?

How to Do Prime Factorization?

To do the prime factorization of a number, we have two popular methods. Let's look at each of them in detail.

Ladder Method

Example

Find the prime factorization of $\hspace{0.2em} 12 \hspace{0.2em}$ .

Solution

Step 1. Write the number with an L-shape around it.

Step 2. Think of a prime number that would divide into $\hspace{0.2em} 12 \hspace{0.2em}$ evenly. Write that factor on the left of the L-shape and the result of division (the quotient) below.

In this case, the factor is $\hspace{0.2em} 2 \hspace{0.2em}$ , and dividing $\hspace{0.2em} 12 \hspace{0.2em}$ by $\hspace{0.2em} 2 \hspace{0.2em}$ , we get $\hspace{0.2em} 6 \hspace{0.2em}$ .

Step 3. Repeat step $\hspace{0.2em} 2 \hspace{0.2em}$ with the new number ( $\hspace{0.2em} 6 \hspace{0.2em}$ ) and keep going until you get $\hspace{0.2em} 1 \hspace{0.2em}$ at the bottom.

Step 4. That's it. On the left of the ladder, we have the prime factors that, when multiplied together, give $\hspace{0.2em} 12 \hspace{0.2em}$ .

So, the prime factorization of $\hspace{0.2em} 12 \hspace{0.2em}$ is -

12 = {\color{Red} 2} \times {\color{Red} 2} \times {\color{Red} 3}

Example

Find the prime factorization of $\hspace{0.2em} 60 \hspace{0.2em}$ .

Solution

Just like we did in the previous example, let's create the prime factor ladder for $\hspace{0.2em} 60 \hspace{0.2em}$ .

So the required prime factorization is -

60 = {\color{Red} 2} \times {\color{Red} 2} \times {\color{Red} 3} \times {\color{Red} 5}

Factor Tree Method

Example

Find the prime factorization of $\hspace{0.2em} 60 \hspace{0.2em}$ .

Solution

Step 1. Split the given number into two factors as shown below. It's best if at least one of the factors is a prime number. But not necessary, as you can see in this case.

Step 2. Repeat the process with the resulting numbers until you can't split them any further.

I have used the red color to indicate that the number is prime and so cannot be split into smaller factors.

Step 3. When you can't split the numbers any further, it means the numbers at the free end of each branch (in red), are all prime. And if you multiply them together, you will get the number at the top.

So the prime factorization of $\hspace{0.2em} 60 \hspace{0.2em}$ is -

60 = {\color{Red} 2} \times {\color{Red} 2} \times {\color{Red} 3} \times {\color{Red} 5}

Example

Find the prime factorization of $\hspace{0.2em} 84 \hspace{0.2em}$ .

Solution

Let's create the factor tree, as we did for the previous example.

Hence, the prime factorization of $\hspace{0.2em} 84 \hspace{0.2em}$ -

84 = {\color{Red} 2} \times {\color{Red} 2} \times {\color{Red} 3} \times {\color{Red} 7}

Note - Depending on how you split the numbers into factor pairs, you can get different factor trees for the same number. That's fine. In the end, the prime factors (the numbers at the free ends) will always be the same.

For example, here are two alternative factor trees for 84. See how all three factor-trees (two below and one above) give the same prime factorization in the end.

No. Factors can only be integers. Factors cannot be fractions or decimals.

Yes. Any integer multiplied by $\hspace{0.2em} 0 \hspace{0.2em}$ gives $\hspace{0.2em} 0 \hspace{0.2em}$ as the product. Hence every integer is a factor of $\hspace{0.2em} 0 \hspace{0.2em}$ .

No. $\hspace{0.2em} 0 \hspace{0.2em}$ multiplied by anything gives $\hspace{0.2em} 0 \hspace{0.2em}$ and nothing else. So, $\hspace{0.2em} 0 \hspace{0.2em}$ cannot be a factor of any non-zero number.

Yes, as I mentioned earlier, factors can be negative too. However, generally, when we say factors, we mean positive factors.

And that brings us to the end of this tutorial on factors and how to do prime factorization. Until next time.

Factors and Prime Factorization

What Are Factors?

Prime Factorization

How to Do Prime Factorization?

Ladder Method

Factor Tree Method