Factors

You and your friends are going to share a batch of cupcakes. There are 24 cupcakes, but you don’t know how many of your friends are going to show up to your party. Of course, everyone may not want several cupcakes, but your friends are picky about their cupcakes. You need to make sure that there are enough cupcakes that everyone will have an equal number of cupcakes. Then everyone will be happy. This is where factors come into play. You need to know if you need to make more cupcakes depending upon the number of people at your party or secretly eat some of the cupcakes before your friends arrive. Nine of your friends RSVP and tell you they are coming. Including yourself, that will mean 10 people will be at the party.

However, that will not work with the number of cupcakes that you have. Ten is not a factor of 24. You will need to bake 6 more cupcakes, so everyone will have an equal share, and you won’t have to worry about a fight taking place over the last cupcake.

What Are Factors?

Probably without realizing it, you use factoring frequently during your day. Factoring is not a skill like logarithms and geometric proofs that you will rarely ever use again outside of your high school math class. Instead, this is a math skill you probably use every day…and without even knowing it. So, needless to say, it is an important skill!

A factor is a number or even a set of numbers that divide evenly into a larger number. Let’s take a large round number that would be easy to work with…100. The number 100 has several factors. This means that several numbers from 1 to 100 can evenly divide into 100 without having any remainders. If you look below, the chart shows a list of large numbers and their factors. See if you can explain why these are factors of the large numbers.

NUMBER	FACTORS
3	1, 3
18	1, 2, 3, 6, 9, 18
50	1, 2, 5, 10, 25, 50
88	1, 2, 4, 8, 11, 22, 44, 88
100	1, 2, 4, 5, 10, 20, 25, 50, 100

Let’s first look at 100. The number 100 has 9 factors. Each of these numbers can factor into 100 evenly. You may notice that these factors on the left are smaller numbers and actually can be multiplied with a larger number on the right to equal one hundred. For example,

$$1\;\times\;100\;=\;100$$,

$$2\;\times\;50\;=\;100$$,

$$4\;\times\;25\;=\;100$$ and

$$5\;\times\;20\;=\;100$$. Now, 10 doesn’t have a partner to equal 100. It is multiplied by itself, so

$$10\;\times\;10\;=\;100$$. Each factor has a partner, but keep in mind that the factor may end up partnering with itself to make the large number.

With the number 3, you may see that it has only 2 factors. This is not because 3 is such a small number, but 1 and itself are the only numbers evenly divided into 3. There are other numbers that only have two factors like the number 3. A number with even three digits only has 2 factors. One example is 311. Its only factors are 1 and itself, 311. These are the only divisible numbers for 311 with having a remainder.

How Does Factoring Work?

Now that you know what factoring is you can now learn how to factor a large number. Basically, you are finding the factors for a number. That is easier said than done. Let’s look at the steps to do this.

STEPS

Step 1: Look at the large number.

Step 2: Think of a number that can evenly be divided into that number.

Step 3: Then find its factor partner as if they were married.

Step 4: Now find all of the factors and their partners.

Step 5: Look at each factor’s partner, and make sure that if multiplied together, it would equal to the large number, 75.

So, 1 times 75 equals 75, 3 times 25 equals 75, and 5 times 15 equals 75. All good there!

This looks easy? Right? Well, not so fast. The problem most people have is that they do not find all of the factors. This part can be tricky. Let’s look at the example. Find the factors for 75. Obviously, all positive numbers, like 75, have two factors which are 1 and itself. Therefore, 1 and 75 are factors for 75, but there are more. Five can also be divided into 75 evenly. What is 5’s factor partner? In other words, what number can be multiplied by 5 to get 75? It is 15. So, 15 is a factor. There are two more numbers that can be multiplied together to equal 75. That’s right! Three and 25 can be multiplied together to make 75.

TIP: How do you know that these are all of the factors? Here is a trick that will help you make sure that you have all of the factors for this large number. Let’s look at an example to show how this trick works. Let’s take the number 120. We need to find all of the factors for 120.

So, the best way to do this is to start with 1 and the number itself. Write these two numbers on either side of the page as shown below. We can’t draw blanks for each factor because we don’t know if this number will have 2 factors or 102 factors or any amount in between

Now, start using numbers 2 – 10. Take each number and see if it divides evenly into 120. Two divides evenly into 120. I need its factor partner which would be 60 because 2 x 60 equals 120. Now, write in 2 and 60.

Let’s take 3. Does 3 divide evenly into 120? Yes, it does. Three times 40 equals 120. Write those in.

Does 4 divide evenly into 120? Yes, and its factor partner is 30. Write those in.

As you can see, I don’t have a lot of room between 4 and 30. That’s okay. I can either rewrite those numbers so that they are smaller, or I can squeeze the rest into that small space. Neatness really doesn’t matter as long as you can read all of the numbers.

Now, 5 also divides evenly into 120 as well as 6. Their partners are 24 and 20. Let’s add those to our line of factors.

At this point, I know that I only have to focus on the middle numbers that are between 6 and 20. This means that the only possible factors and their partners remaining are between 6 and 20. Let’s check 7. Seven does not divide evenly into 120 because there is a remainder. However, 8 does, and its partner is 15. Let’s squeeze those into our line of numbers.

Now, you only have to worry about any factors and their partners between 8 and 15. Check out 9. Nope, there is a remainder with 9. How about 10? You probably won’t need to use a calculator for this one because

$$10\;\times\;12$$ equals 120.

The only number left would be 11, but 11 has a remainder if we divide it into 120. If we wanted to clean up our factor line a bit or if you have trouble reading the numbers, you can rewrite them as such.

With so many factors, it is difficult to know how small to write your numbers, so do your best and remember this is not a neatness contest. It is okay to be a little messy. Your goal is to find all of the factors.

Example 1:

Let’s try an example like 256. What are its factors?

Step 1: Look at the number: 256.

Step 2: Think of a number that can evenly be divided into that number.

Step 3: Then find its factor partner as if they were married.

Step 4: Now find all of the factors and their partners. Use the tip from above.

Step 5: Look at each factor’s partner, and make sure that if multiplied together, it would equal to the large number, 256.

Now that you have finished, how many factors does 256 have? Did you find that 256 has 9 factors?

Let’s look at these factors and match them with their factor partners.

The number 16 can be multiplied by itself to get 256. So, at this point, there are no other factors because there is not a whole number between 16 and 16.

Using Factors in Math and Everyday Life

You may remember that I told you that factoring is used all the time, so let’s look at these ways. First, obviously, you will use factoring in your math class. You may have to do this or find one factor for a number when calculating more complicated math problems.

When you are dividing, you may need to find the factor that goes into that dividend evenly. With certain division word problems, they require a whole number as the answer. For example, if the word problem was including the number of marbles, people, or dogs, the answer would need to be a whole number. A marble would most likely not be divided since it is so difficult to break. Look at the word problem below.

From this word problem, you may be able to tell that this should be a whole number because you would not cut a dog in half. Oh no!

Can you imagine? Poor Fido!

Instead, when you read a certain word problem and you are dividing, you may realize that you need to know the factors for that number because there will not be a remainder. In this example, five dogs will be placed in each doghouse. There could not be a fraction of a dog.

Ways that you use factoring in your everyday lives would be for dividing food into equal amounts, comparing prices with an item, and making the change or dividing money into equal amounts.

CHALLENGE QUESTIONS