Finding Factors of a Number — Theory & Examples

Listing Factors of a Number

Let's begin this topic with a question.

Is every whole number actually the product of two numbers?

The answer is yes. These numbers that are multiplied together to create another number are called factors. Every number—even prime numbers—have factors, which means they can all be factored.

Whenever we talk about factoring a number, we are talking about breaking the number apart into smaller numbers. We factor a number, or break the number apart, by finding all of the numbers that can be multiplied together to get that number.

Each number has at least two factors, itself and $1$.

When listing factors of a number, I like to use something called a "factor rainbow." Factor rainbows help us find all of the factors of a number while also keeping track of "factor pairs." Factor pairs are sets of two numbers that are multiplied together to get the target number that we are factoring.

Begin with $1$ and the number itself. For our example we will start with the number $36$.

After listing $1$ and the number itself, count up from one to find the other factors. So the next number to check is $2$. In this case $2$ can be multiplied by $18$ to get $36$ so $2$ and $18$ are a factor pair.

The next number we can check is $3$. $$3\times12=36$$ So $3$ and $12$ are a factor pair.

Next we can check $4$. $$4\times9=36$$ So $4$ and $9$ are a factor pair.

Next we can check $5$. $5$ can not be multiplied by a whole number to get $36$ therefore it is not a factor. The next number we can check is $6$. $$6\times6=36$$ So $6$ is a factor of $36$.

Next step would be to check $7$ and $8$. There is no way to multiply $7$ or $8$ by a whole number to get $36$ therefore neither of them are factors of $36$.

The next number to check is $9$, but as we can see in the factor rainbow above, $9$ is already listed as a factor. Since we have now reached a number that is already listed in the rainbow, we know we have found all of the factors of $36$.

Here's another example, where we find the factors of $72.$

One disclaimer though. This is different from finding the prime factors of a number. That is called a prime factorization.

Bonus: Factoring isn't just for integers! You can also factor monomials!

Finding a factor can seem difficult at first, but there are techniques to make it easier. Usually, finding factors is done through a guess-and-check method, but there are different rules that make finding a factor easier.

$1$ is a factor of everything!

For every whole number, $1$ is a factor. This is because for any whole number, it is true that: $$\text{number} \times 1 = \text{number}$$ This is called the identity property of multiplication. In addition, this means that any number is a factor of itself. That's $2$ factors right off the bat!

Since any number has factors of $1$ and itself, they will have $2$ to start, before you even have to think about it!

Even numbers have a factor of $2$!

Even numbers are always divisible by $2$, so that means that $2$ is always a factor of any even number, along with half of that even number. Remember, a number is even if it ends in a $0$, $2$, $4$, $6$, or an $8$. Let's do another example:

Each even number in that list ($6$, $12$, $14$, $20$, and $28$) has a factor of $2$, and a factor of half of those numbers ($3$, $6$, $7$, $10$, and $14$, respectively)

If the sum of the digits in a number is divisible by $3$, so too is the number!

This is one of the harder divisibility rules to remember and apply. It states that if the sum of digits in a number is divisible by $3$, then so too is the original number. For example, let's see if $16$ or $18$ work: $$16: 1 + 6 = 7$$ Since $7$ isn't divisible by $3$, we can conclude that $16$ is not either. Indeed, if we check we see that $16 \div 3 = 5.33333...$ $$18: 1 + 8 = 9$$ Since $9$ is divisible by $3$, we know that so is $18$. Checking, we get that $18 \div 3 = 6$. Let's do another example:

In order: $$1 + 9 = 10$$ $$2 + 1 = 3$$ $$2 + 4 = 6$$ $$2 + 5= 7$$ $$2 + 6 = 8$$ $$2 + 9 = 11$$ $$3 + 0 = 3$$ $$3 + 3 = 6$$ Therefore, since $3$ and $6$ are divisible by $3$ the correct answers are: $21$, $24$, $30$, and $33$.

If a number ends in a $0$ or a $5$, it is divisible by $5$!

The final rule of thumb I'll go over here involves numbers that are divisible by $5$. Any number that ends in a $0$ or a $5$ is divisible by $5$, and all other numbers are not. For example, $10$, $15$, $25$, and $2005$ are all divisible by $5$, but $11$, $14$, $23$, and $2016$ are not. Another example:

Since $85$, $145$, $2{,}010$ and $38{,}955$ all end in a $5$ or a $0$, they are the correct answers here.

Check each number until you get to a factor you've already seen!

After these rules of thumb, there is not much else to do but to check each number to see if they have factors. However, there are signs that tell you that you can stop. The one that I use the most is the following: if, while checking for factors, you reach a number that you already listed as a factor, you can stop.

Let's use the example of $12$. First, we know that: $$1 \times 12 = 12$$ So $1$ and $12$ are factors. Then since $12$ is even, we know: $$2 \times 6 = 12$$ So $2$ and $6$ are also factors. Then since $1 + 2 = 3$, we know that $12$ is divisible by $3$: $$3 \times 4 = 12$$ So $3$ and $4$ are factors. After this, we would normally continue by checking if $12$ is divisible by $4$. But since we already know that $4$ is a factor alongside $3$, we are guaranteed to not find any new factors after this. Therefore, we know that the factors of $12$ are $1$, $2$, $3$, $4$, $6$ and $12$.

Let's do one final example:

Proceeding as shown before, we know right off the bat that: $$1 \times 15 = 15$$ So, $1$ and $15$ are also factors.

Since $15$ is odd, we know that $2$ is not a factor.

Since $1 + 5 = 6$ we know that $15$ is divisible by $3$, so after checking we get: $$3 \times 5 = 15$$ So, $3$ and $5$ are factors.

$15 \div 4 = 3.75$, so $4$ is not a factor.

Then, normally, we would proceed to check $5$, but as we already know that $5$ is a factor, now we can be done. Therefore, the correct answer is $1,3,5,15$.

Through a combination of rules of thumb and some guess-and-check, finding factors can be quick and easy!

**(Video) Finding Factors of a Number **

by Don't Memorize

Here is a video by Don’t Memorize about finding all the factors of a number.

$\large{\text{Summary}}$

As the video said, to find the factors of a number, we first need to write the number as a product of two positive whole numbers. We have to think of all the possible combinations of numbers.

Always start off with $1$ as $1$ will be a factor of every whole number. Next, see if the number is divisible by $2$, which is the next whole number.

Now, move on to the next whole number, $3$, and so on, until all the factors are considered for.

Factors are essentially exact divisors of that number.