# Prime and Composite Numbers

Hello, and welcome to this video about **prime and composite numbers**! Today we’ll learn what prime and composite numbers are, how to classify numbers as prime or composite, and why this is important.

Before we get started, let’s review a few things. First, **factors** are numbers that you multiply together to get another number. For example, in the equation \(2\times{3}=6\), 2 and 3 are factors of 6. \(1\times{6}\) also equals 6, so 1 and 6 are factors as well. The answer to a multiplication problem is called the **product**. In the examples given, 6 is the product of \(2\times{3}\) and \(1\times{6}\).

A **prime number** is a number greater than 1 that is not the product of two smaller numbers. The only factors of a prime number are 1 and the number itself. For instance, 13 is a prime number because the only factors of 13 are 1 and 13. In other words, the only way to make a product of 13 is by multiplying \(1\times{13}\).

In contrast, a **composite number** is a positive number that is the product of two smaller numbers. A composite number can have many factors. For example, 12 is a composite number because it has several factors: \(1\times{12}\), \(2\times{6}\), and \(3\times{4}\). Since there are many ways to multiply to get a product of 12, it is classified as a composite number.

All positive integers are either prime or composite. To determine whether a number is prime or composite, simply list all the factors of that number. In other words, list all the ways that you can multiply to get the number in question as the product. If the number has many factors, then it is composite. If the only factors are 1 and the number itself, then it’s prime.

Let’s take a look at an example together. Determine whether 24 is a prime or composite number. To do so, make a list of all the factors of 24. Remember, factors are numbers that are multiplied together to get a product. In this case, the product is 24.

Start with 1 and work your way up from there. Ask yourself, “1 times what number equals 24?” \(1\times {24}=24\), so 1 and 24 are factors. Next is 2. 2 times what number equals 24? \(2\times{12}=24\), so 2 and 12 are also factors of 24. After 2 is 3. 3 times what number equals 24? \(3\times{8}=24\), so 3 and 8 are factors of 24 as well. After 3 comes 4. 4 times what number equals 24? \(4\times{6}=24\), so 4 and 6 are factors of 24.

Now let’s move on to 5. 5 times what number equals 24? Since there isn’t a number we can multiply by 5 to get a product of 24, 5 is not a factor. Therefore, we can skip 5 and move on to 6. 6 times what equals 24? Well, \(6\times{4}=24\), so 6 and 4 are factors of 24. But notice that we’ve already named these numbers as factors of 24, since \(4\times{6}=24\). Repeating factors indicate there are no more factors of 24, we’ve already named them all.

\(2\times{12}=24\)

\(3\times{8}=24\)

\(4\times{6}=24\)

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Since there are several factors of 24, this is a composite number.

Let’s take a look at another example together. Determine whether 11 is prime or composite. To do so, make a list of all the factors of 11. Remember, these are all the numbers we can multiply to get a product of 11.

Let’s start with 1. Ask yourself, “1 times what number equals 11?” \(1\times{11}=11\), so 1 and 11 are factors. Next is 2. Nothing times 2 is 11, so we can skip 2 and move onto 3. You can’t multiply anything by 3 to get 11, so move onto 4. Again, nothing times 4 equals 11, so 4 is not a factor. Let’s move onto 5. 5 times what number equals 11? Since there isn’t a number we can multiply by 5 to get to 11, 5 is not a factor either. What about 6? Well, \(6\times{1}=6\), and \(6\times{2}=12\). So there aren’t any numbers we can multiply by 6 to get to 11.

7, 8, 9, and 10 aren’t factors of 11 either. 11 is a factor because \(1\times{11}=11\). However, we’ve already listed 1 and 11 as factors. Since these are the only two factors of 11, that means it’s a prime number because the only way to get a product of 11 is by multiplying it by 1 and 11.

Now it’s your turn. List the factors of 18 to determine whether it’s prime or composite. Pause the video here and try this one yourself. When you’re done, resume the video and we’ll go over it together.

To determine whether 18 is a prime or composite number, make a list of all the factors of 18. Remember, these are all the numbers we can multiply to get a product of 18.

Start with 1. Ask yourself, “1 times what number equals 18?” \(1\times{18}=18\), so 1 and 18 are factors. Next is 2. \(2\times{9}=18\), so 2 and 9 are factors. What about 3? Well, \(3\times{6}=18\), so 3 and 6 are factors.

Next comes 4. 4 times what number equals 18? Since there isn’t a number we can multiply by 4 to get to 18, 4 is not a factor. Skip it and move onto the next number, which is 5. Nothing time 5 is 18, so 5 is also not a factor. Skip it and move onto the next number, which is 6. \(6\times{3}=18\), so 6 and 3 are factors. But we’ve already named these numbers as factors of 18, because \(3\times{6}=18\). Repeating factors indicate that there are no more factors of 18, and we have already named them all.

\(2\times{9}=18\)

\(3\times{6}=18\)

So the factors of 18 are 1, 2, 3, 6, 9, and 18. Since there are several factors of 18, we can classify it as a composite number.

Great job!

Many patterns and number concepts rely on prime numbers, as well as expressing composite numbers as the product of their prime numbers. For instance, computer encryption relies on prime numbers and prime factorization. Factoring large numbers can take a really long time, even for computers. As a result, prime numbers and factorization are used for encryption purposes to protect confidential data. Some examples involving encryption include logging into your bank account’s website, inputting your credit card information to make an online purchase, and sending a confidential email.

Let’s take a look at one more example together.

*The area of a rectangular patio is 39 square feet. What are the possible whole number dimensions of the patio?*

To find the area of a rectangle, multiply length times width. In this case, we know that the area is 39 square feet. What numbers can we multiply together to get a product of 39? To answer this question, make a list of all the factors of 39.

Let’s start with 1. Ask yourself, “1 times what number equals 39?” \(1\times{39}=39\), so 1 and 39 are factors. Next comes 2. 2 times what equals 39? Well, there isn’t a number we can multiply by 2 to get to 39, so 2 is not a factor. We skip it and move onto the next number, which is 3. \(3\times{13}=39\), so 3 and 13 are factors. Working our way through the next several numbers, there are no other factors of 39.

\(3\times{13}=39\)

Therefore, the possible dimensions of a patio with an area of 39 square feet are: 1 foot by 39 feet and 3 feet by 13 feet.

Since there are several factors of 39, we can classify it as a composite number.

## Review

I have one more problem for you to try. This one’s a little more difficult, but I know you can handle it. Take a look at the problem, pause the video, and try it yourself. When you’re done, resume the video and we’ll go over everything together.

*The area of a rectangular shed is 84 square feet. List all possible whole number dimensions of the shed. Is the area of the shed a prime or composite number? How do you know?*

Pause the video here and see if you can answer these questions yourself.

To find the area of our rectangle, multiply the length times the width. In this case, we know that the area is 84 square feet. So what numbers can we multiply together to get a product of 84? To answer this question, we just have to make a list of all the factors of 84.

Here are the factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

\(2\times{42}=84\)

\(3\times{28}=84\)

\(4\times{21}=84\)

\(6\times{14}=84\)

\(7\times{12}=84\)

Therefore, the possible dimensions of a garden with an area of 84 square feet are: 1 foot by 84 feet, 2 feet by 42 feet, 3 feet by 28 feet, 4 feet by 21 feet, 6 feet by 14 feet, and 7 feet by 12 feet.

Since there are several factors of 84, we can classify it as a composite number.

Nice work!

I hope this video about prime and composite numbers was helpful. Thanks for watching, and happy studying!

## Practice Questions

**Question #1:**

List all the factors of \(7\), then determine whether \(7\) is a prime or composite number.

\(1\), \(2\), \(5\), \(7\); composite

\(1\), \(3\), \(7\); composite

\(1\), \(6\); prime

\(1\), \(7\); prime

**Answer:**

To find all the factors of \(7\), start with \(1\) and ask, “\(1\) times what number equals \(7\)?”

\(1\times7=7\)

So \(1\) and \(7\) are both factors of \(7\).

Next, ask “\(2\) times what number equals \(7\)?” Well, no whole numbers give us seven when multiplied by \(2\), so \(2\) is not a factor of \(7\).

Next, move on to \(3\). “\(3\) times what number equals \(7\)?” Again, no number multiplied with three equals \(7\). This means that \(3\) is also not a factor of \(7\).

Continue asking this question with each number until \(7\). Just like \(2\) and \(3\), the numbers \(4\), \(5\), and \(6\) are also not factors of \(7\) because no numbers will multiply with them to equal \(7\).

Finally, ask “\(7\) times what number equals \(7\)?” Clearly, \(7\times1=7\), so \(7\) and \(1\) are factors of seven. We already knew that from the first equation, though. This means that \(1\) and \(7\) are the only two factors of \(7\). A prime number is a number whose only factors are \(1\) and itself, so \(7\) is a prime number.

**Question #2:**

List all of the factors of \(16\), then determine whether it is prime or composite.

\(1\), \(2\), \(4\), \(8\), \(16\); composite

\(1\), \(16\); composite

\(1\), \(16\); prime

\(1\), \(4\), \(16\); composite

**Answer:**

To find all factors of \(16\), start with \(1\). Ask “\(1\) multiplied by what number is \(16\)?” We know that \(1\times16=16\), so \(1\) and \(16\) are both factors of \(16\).

Next, ask “\(2\) multiplied with what number equals \(16\)?” We know that \(2\times8 =16\), so both \(2\) and \(8\) are factors of \(16\).

Moving along, consider \(3\). Ask “\(3\) multiplied with what number equals \(16\)?” Since \(16\) is not divisible by three, \(3\) is not a factor of \(16\).

Continue to \(4\). What number, multiplied by \(4\), equals \(16\)? Four! \(4\times4=16\). So \(4\) is a factor of \(16\).

Keep asking these questions for the next numbers: \(5\), \(6\), and \(7\). You’ll see that they are not factors of \(16\).

Now check \(8\). What number, multiplied by \(8\), equals \(16\)? Two. \(8\times2=16\). But since we already covered \(8\) and \(2\), we know we are done looking for factors now.

From our work, we know that \(1\), \(2\), \(4\), \(8\), and \(16\) are all of the factors of \(16\). Since there are five numbers rather than just two (say, \(1\) and \(16\)), the number \(16\) is composite.

**Question #3:**

List all of the factors of \(17\), then determine whether it is a prime number or a composite number.

\(1\), \(4\), \(5\), \(17\); composite

\(1\), \(17\); composite

\(1\), \(17\); prime

\(1\), \(3\), \(7\), \(17\); composite

**Answer:**

To determine this solution, you must go through each number \(1\) through \(17\) and ask:

“\(1\) times what number equals \(17\)?” \(1\times17=17\), so \(1\) and \(17\) are factors of \(17\).

“\(2\) times what number equals \(17\)?” No number, multiplied by \(2\), equals \(17\). \(2\) is not a factor.

“\(3\) times what number equals \(17\)?” No number, multiplied by \(3\), equals \(17\). \(3\) is not a factor.

For the numbers \(2-16\), none are factors of \(17\). For \(17\), clearly \(17\times1=17\), but we already know that \(1\) and \(17\) are factors of \(17\).

From these sequential questions, it can be determined that the only factors of \(17\) are \(1\) and \(17\) itself. From what we know about classifying prime numbers, this means that \(17\) is prime.

**Question #4:**

Addie has \(27\) pencils she wants to give away to her friends, and she is considering how she can group the pencils so that each recipient will get an equal amount. How can Addie divide up the pencils evenly?

\(27\) groups of \(1\) pencil each

\(27\) groups of \(1\) pencil each, or \(7\) groups of \(2\) pencils each

\(27\) groups of \(1\) pencil each, or \(9\) groups of \(3\) pencils each, or \(7\) groups of \(2\) pencils each

\(27\) groups of \(1\) pencil each, or \(9\) groups of \(3\) pencils each

**Answer:**

This is a factorization question in disguise. You must find all the factors of \(27\).

Because all numbers have \(1\) and themselves as factors, you probably already know that \(1\) and \(27\) are factors of \(27\).

Next, is \(2\) a factor of \(27\)? Well, no number can be multiplied with \(2\) to get \(27\), so \(2\) is not a factor.

What about \(3\)? Since \(3\times9=27\), both \(3\) and \(9\) are factors of \(27\).

Continue with the numbers \(4\) through \(26\). You will see that there are no other factors in this range, aside from \(9\), which was just covered.

Since \(1\times27=27\) and \(3\times9=27\), this means that Addie can group the pencils in \(27\) groups of \(1\) pencil (one pencil for each of \(27\) friends) or in \(9\) groups of \(3\) pencils (three pencils for each of nine friends).

**Question #5:**

Harold has \(49\) cans of beans, and he wants to stack them so that every stack is the same height. Find all such ways the beans can be arranged.

\(1\) stack of \(49\) cans, or \(5\) stacks of \(8\) cans each, or \(49\) stacks of \(1\) can each

\(1\) stack of \(49\) cans, or \(5\) stacks of \(9\) cans each, or \(49\) stacks of \(\)1 can each

\(1\) stack of \(49\) cans, or \(7\) stacks of \(7\) cans each, or \(49\) stacks of \(1\) can each

\(1\) stack of \(49\) cans, or \(49\) stacks of \(1\) can each

**Answer:**

This is another factorization problem. To determine the solution, find all factors of \(49\). Clearly, \(1\) and \(49\) are both factors because \(1\times49=49\).

Next, notice that \(49\) is not an even number, so \(2\) cannot be a factor.

\(49\) is also not divisible by \(3\), so \(3\) cannot be a factor either.

Similarly, the numbers \(4\), \(5\), and \(6\) are also not factors of \(49\) because no number multiplies with them to get \(49\).

Consider \(7\) though. What number multiplied by \(7\) equals \(49\)? \(7\times7=49\). This means that \(7\) is a factor of \(49\). And because \(7\) multiplies with itself to get \(49\), we can stop here. All further numbers will not provide unique factors.

The factors of \(49\) are \(1\), \(7\), and \(49\). This means that Harold can stack the cans in one very tall stack of \(49\) cans, in \(7\) stacks of \(7\) cans each, or in \(49\) “stacks” of one can each.