Alternative Format

Try This

Building Blocks

Many things in our daily lives are created by combining simple things together to form more complex things.

Source: Ingredients - didecs/iStock/Getty Images

Source: Cogito Ergo Sum - Descartes, R. (1637). Discourse on the Method.

Source: H₂0 - Chromatos/iStock/Getty Images; Splash - BlackJack3D/E+/Getty Images

In math, we often think numbers are independent of each other. However, numbers are also made up of building blocks in much the same way as a pie is made up of ingredients.

Lesson Goals

• Define prime and composite numbers.
• Determine the prime factorization of a number using factor trees.
• Express the prime factorization of a number using exponents.

Try This!

What do you notice about the patterns created on the chart?

Why are some numbers highlighted?

Why are some numbers circled?

Think about this problem, then move on to the next part of the lesson.

Source: Sieve - Ricordisamoa. (2007, November 7). Sieve of Eratosthenes animation. Retrieved from Wikimedia Commons.
Used under CC BY-SA 3.0.

Prime and Composite Numbers

Factors Review

Example

The factors of \(12\) are \(1\), \(2\), \(3\), \(4\), \(6\), and \(12\).

How did we find these factors? When we are looking for factors that divide evenly into a number, we often consider what pairs of numbers multiply together to give us that number. For example,

\(3 \times 4 = 12\)

\(3\) and \(4\) are factors of \(12\)

Prime and Composite Numbers

When we consider the factors of \(13\), we find that there are only two of them: \(1\) and \(13\). No other integers will divide evenly into \(13\).

This is quite the contrast to the factors of \(12\), where there were a total of six factors. Recall the factors of \(12\) are \(1\), \(2\), \(3\), \(4\), \(6\), and \(12\).

A number, like \(13\), that only has two factors, \(1\) and itself, is called a prime number.

\(12\) is clearly not a prime number because it has more than two factors. A number like \(12\) is called a composite number.

Let's look at a few examples of both prime and composite numbers.

Example 1

Solution

To decide whether \(27\) is a prime or composite number, we can consider whether \(27\) has exactly two factors or more than two factors.

Some factors of \(27\) include:

So we want to consider if there is any other number that divides evenly into \(27\) other than \(1\) and \(27\), itself.

This means \(3\) is also a factor of \(27\).

Therefore, \(27\) is a composite number because it has more than two factors.

In fact, \(27\) has four total factors. The factors of \(27\) are \(1\), \(3\), \(9\), and \(27\).

Example 2

Solution

We can consider whether \(11\) has any factors other than \(1\) and \(11\) itself.

If it doesn't, it will be a prime number. And if it does, it will be a composite number.

It is sometimes easiest to start at \(2\) and work our way up logically removing integers that do not divide into \(11\). \(2\) does not divide evenly into \(11\) since \(2\) only divides into even numbers.

\(2\) ✘

This is the same for all even numbers, so \(4\), \(6\), \(8\), and \(10\) will not divide evenly into \(11\) either.

Continuing with the odd numbers, neither \(3\) nor \(5\) divide evenly into \(11\).

\(7\) is more than half of \(11\), so \(7\) is too large to divide evenly into \(11\). We can conclude that any number larger than \(7\) will also not divide evenly into \(11\).

The only factors of \(11\) are \(1\) and \(11\).

Therefore, \(11\) is a prime number because it only has two factors, \(1\) and itself.

Try This Problem Revisited

Now that we are more comfortable with the prime and composite numbers, let's have a look at the Try This problem from earlier in the lesson.

You may have noticed some patterns in the chart. Here's how it works.

When complete, all of the composite numbers are highlighted since they are all multiples of other numbers. All of the circled numbers are prime numbers.

Image Description

The prime numbers less than \(120\) include

\(\begin{align*}& 2,~3,~5,~7,~11,~13,~17,~19,~23,~29,~31,~37,~41,~43,~47,~53,~59,\\ & 61,~67,~71,~73,~79,~83,~89,~97,~101,~103,~107,~109,~113\end{align*}\)

Source: Sieve - Ricordisamoa. (2007, November 7). Sieve of Eratosthenes animation. Retrieved from Wikimedia Commons.
Used under CC BY-SA 3.0.

Visualizing this process helps us to understand the differences between prime numbers and composite numbers. Since we'll be using prime numbers in future lessons, it is helpful to familiarize yourself with some of the prime numbers, especially those that are less than \(30\).

The Sieve of Eratosthenes

Eratosthenes was a Greek mathematician and all around scholar who, 2200 years ago, created the Sieve of Eratosthenes.

The Sieve shows all prime and composite numbers up to a given point. For the Sieve that we have here, we stop at \(120\).

Source: Sieve - Ricordisamoa. (2007, November 7). Sieve of Eratosthenes animation. Retrieved from Wikimedia Commons.
Used under CC BY-SA 3.0.

Check Your Understanding 1

Question

For each number below, determine whether it is a prime number or a composite number.

1. \(10\)
2. \(49\)
3. \(7\)
4. \(11\)

Answer

1. \(10\) is composite.
2. \(49\) is composite.
3. \(7\) is prime.
4. \(11\) is prime.

Feedback

Prime number numbers have only two factors, \(1\) and the number itself. Composite numbers have more than two factors. Listing the factors of each number gives us the following:

Number	Factors
\(10\)	\(1,~2,~5,~10\)
\(49\)	\(1,~7,~49\)
\(7\)	\(1,~7\)
\(11\)	\(1,~11\)

Is \(1\) Prime?

You may have been wondering about the number \(1\). Is \(1\) a prime number? Based on our definition of a prime number, \(1\) seems like it would fit, since \(1\) can be divided evenly by \(1\) and itself.

The number \(1\) has only one factor. So it doesn't fit the second part of our definition — that a prime number has exactly two factors.

Therefore, \(1\) is not considered to be a prime number.

Prime Factorization

Introduction

Here's a special diagram for the number \(36\).

What rules do you think were used to create this diagram?

When you have an idea, move on to the Explore This activity to see if your idea works for other numbers as well.

Explore This 1

Description

Notice the number at the top. What do you notice about the resulting diagram? What rules do you think are used to create this diagram?

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/YBtVdgsq

Online Version

https://ggbm.at/YBtVdgsq

Factor Trees

During the Explore This activity, you may have noticed that each number in the diagram is the product of the two numbers below it.

For this reason, these diagrams are called factor trees. Factor trees begin at the top with a number, and then branch downward to two numbers that multiply to equal the top number.

Adding this branch wouldn't really tell us any more information. So the branches of factor trees end when we reach prime numbers.

A factor tree is a diagram that identifies the prime factors and how many times each prime factor appears for a specific number. 

These prime factors are like the building blocks that combine together to create the number at the top of the factor tree.

Prime Factorization

Let's create the factor tree of \(12\).

The product of the prime numbers that are on the end of the branches is called the prime factorization of \(12\).

We write the prime factorization of \(12\) as the following multiplication equation:

\(12 = 2 \times 2 \times 3\)

The factor tree is a tool we can use to help us find the prime factorization of a number. To start this factor tree, we chose the factors \(2\) and \(6\). What would happen if, instead, we chose the factors \(3\) and \(4\), since they also multiplied to give us \(12\)?

Let's have a look at that factor tree.

The prime factorization of \(12\):

\(12 = 3 \times 2 \times 2\)

What do you notice about the prime factorizations of the two different factor trees? Both have two \(2\)'s and one \(3\).

When we multiply, the order of the numbers doesn't matter. So both of these prime factorizations give a result of \(12\).

Any factor tree for \(12\) will result in the prime factorization with two \(2\)'s and one \(3\).

It doesn't matter what factors you choose to start with for a factor tree — the prime factorization will always be the same.

Example 3

Solution

To discover the prime factorization of \(26\), we can use a factor tree of \(26\).

\(2\) is a prime number, so we can highlight \(2\).

The prime factorization is

\(26 = 2 \times 13\)

Example 4

Solution

To discover the prime factorization of \(42\), we can use a factor tree of \(42\).

The factor tree is complete, and the highlighted prime numbers will give us the prime factorization of 42:

\(42 = 2 \times 3 \times 7\)

Check Your Understanding 2

Question

Create a factor tree for the following numbers:

1. \(25\)
2. \(37\)
3. \(40\)
4. \(36\)

Answers

1. \(25\)
   
2. \(37\)
   
3. \(40\)
   
4. \(36\)
   

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/ewjA2FWN

Online Version

https://ggbm.at/ewjA2FWN

Prime Factorization Using Exponents

Example 5

Solution

What is the prime factorization of \(32\)? To discover the prime factorization of \(32\), we can create a factor tree of \(32\).

The factor tree is complete, and the highlighted prime numbers will give us the prime factorization of \(32\):

\( 32 = 2 \times 2 \times 2 \times 2 \times 2\)

That's a lot of twos.

Is there an easier way that we could write this repeated multiplication? Of course. We could use exponents.

In this case, the base number will be \(2\). And the exponent will be \(5\). Therefore, the prime factorization of \(32\) is

\(\begin{align*} 32 & = 2 \times 2 \times 2 \times 2 \times 2\\[1ex] 32 & = 2^5 \end{align*}\)

The prime factorization of many numbers will have repeated prime factors. We can use exponents to simplify the expression and make it easier to read.

Example 6

Solution

To discover the prime factorization of \(200\), we can create a factor tree for \(200\).

The factor tree is complete, and the highlighted prime numbers will give us the prime factorization of \(200\) to be

\( 200 = 2 \times 2 \times 2 \times 5 \times 5\)

Since there is repeated multiplication in the prime factorization, we can use exponents to simplify.

\(200 = 2^3 \times 5^2\)

Remember that the exponents \(3\) and \(2\) have a special way that we can say to them. So we could also say this prime factorization as "\(2\) cubed times \(5\) squared."

Check Your Understanding 3

Question

Which of the following represents the prime factorization of \(98\)?

1. \(7^2\times 2\)
2. \(7\times 2\)
3. \(7^2\times 2^2\)
4. \(49\times 2\)

Answer

1. \(7^2\times 2\)

Feedback

Using a factor tree, we can find the prime factorization of \(98\).

\(\begin{align*} 98 & = 7\times 7 \times 2 \\ & = 7^2 \times 2\end{align*}\)

Take It With You

Twin primes are pairs of prime numbers that differ by \(2\).  

The first twin primes are \(3\) and \(5\), since they are both prime numbers and they have a difference of \(2\).

1. How many pairs of twin primes are there below \(100\)?
2. Is the number of pairs of twin primes below \(100\) a prime or a composite number?