# Prime Factorization

## Number written as a product of its prime factors

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Prime Factorization

Connor is working on prime factorization for his math homework. He needs to find the prime numbers that, when multiplied together, produce the number 82. How can Connor complete this problem?

In this concept, you will learn to write the prime factorization of given numbers using a factor tree.

### Prime Factorization Using Factor Trees

When a number is factored, it is broken down into two factors that are either prime numbers or composite numbers. Prime numbers are numbers that have only two factors, one and itself, and composite numbers are numbers that have more than two factors. Some examples of prime numbers are 2, 3, 11, etc. Prime factorization is the process of breaking down a number into a product of all prime numbers.

Here is a composite number.

36

The number 36 can be factored several different ways, but let’s factor it with \begin{align*}6 \times 6\end{align*}.

\begin{align*}36 = 6 \times 6\end{align*}

These two factors are not prime factors. Therefore, both factors can be factored again.

\begin{align*}36 = 6 \times 6 = 2 \times 3 \times 2 \times 3\end{align*}

2 and 3 are both prime numbers.

One way to organize the factors is using a factor tree.

\begin{align*}\begin{array}{rcl} && \quad \qquad \ \ 36\\ && \quad \qquad \big / \quad \big\backslash \\ && \qquad \ \ \ 6 \ \times \ 6 \\ && \qquad \big / \ \ \big\backslash \quad \big / \ \ \big\backslash \\ && \quad \ \ \ 2 \times 3 \ \ 2 \times 3\\ \\ && \ 36 = 2 \times 2 \times 3 \times 3 \end{array}\end{align*}

The number is written at the top of the factor tree. Then it is broken down into a factor pair, \begin{align*}6 \times 6\end{align*}. 6 can further be factored so the factor pairs are written underneath the 6. Each number is continued to be factored until the factors are all prime numbers. Note that 36 is written as a product of its primes at the bottom of the factor tree. Write the 2s together and the 3s together. Grouping like factors will help keep track of them.

The prime factorization is written using exponential notation, a method of writing repeated multiplication.

\begin{align*}36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\end{align*}

The base is the number being repeated and the exponent is the number of times the number is being multiplied. 2 times 2 is written as base 2 with the exponent 2, the number of times 2 is multiplied by itself. 3 times 3 is written as base 3 with an exponent of 2.

### Examples

#### Example 1

Earlier, you were given a problem about Connor’s prime factorization math problem.

Connor needs to find the prime factorization of 82. Use a factor tree to solve this problem.

Then, begin by factoring 82 using factor pairs.

\begin{align*}\begin{array}{rcl} && \qquad \ 82\\ && \qquad \big / \ \ \big\backslash \\ && \quad \ \ 2 \times 41\\ \\ && \ 82 = 2 \times 41 \end{array}\end{align*}

Connor finds that 82 is the product of only 2 prime numbers, 2 and 41.

#### Example 2

Write the prime factorization of 25.

Then, factor 25 into the product of all prime numbers. 25 can be factored into 5 times 5. 5 is a prime number so this tells you that you have reached the bottom of the factor tree.

\begin{align*}\begin{array}{rcl} && \quad \ \ \ 25\\ && \quad \ \ \big / \ \ \big\backslash \\ && \quad \ 5 \times 5 \\ \\ && \ 25 = 5 \times 5 \end{array}\end{align*}

Next, write the factors in exponential notation. The base is 5 and the exponent is 2.

\begin{align*}25 = 5^2\end{align*}

The prime factorization of 25 is \begin{align*}5^2\end{align*}.

#### Example 3

Write the prime factorization for the following number.

\begin{align*}48\end{align*}

Then, factor 48 into the product of all prime numbers.

\begin{align*}\begin{array}{rcl} && \quad \qquad \ 48\\ && \quad\qquad \big / \ \ \big\backslash \\ && \quad\quad \ \ 4 \ \times 12 \\ && \qquad \big / \ \big\backslash \ \ \ \big / \ \ \big\backslash \\ && \quad \ \ 2 \times 2 \ 2 \times 6\\ && \qquad \qquad \quad \ \big / \ \big\backslash\\ && \qquad \qquad \quad 2 \times 3 \\ \\ && 48 = 2 \times 2 \times 2 \times 2 \times 3 \end{array}\end{align*}

Next, write the factors in exponential notation.

\begin{align*}48 = 2^4 \times 3\end{align*}

The prime factorization of 48 is \begin{align*}2^4 \times 3\end{align*}.

#### Example 4

Write the prime factorization for the following number.

\begin{align*}100\end{align*}

Then, factor 100 into the product of all prime numbers.

\begin{align*}\begin{array}{rcl} && \quad \qquad \ 100\\ && \quad\qquad \ \big / \ \ \big\backslash \\ && \quad\quad \ \ \ 2 \times 50 \\ && \qquad \qquad \ \big / \ \big\backslash \\ && \qquad \quad \ \ \ 2 \times 25\\ && \qquad \qquad \quad \ \big / \ \big\backslash\\ && \qquad \quad \quad \quad 5 \times 5\\ \\ && \ 100 = 2 \times 2 \times 5 \times 5 \end{array}\end{align*}

Next, write the factors in exponential notation.

\begin{align*}100 = 2^2 \times 5^2\end{align*}

The prime factorization of 100 is \begin{align*}2^2 \times 5^2\end{align*}.

#### Example 5

Write the prime factorization for the following number.

\begin{align*}144\end{align*}

Then, factor 144 into the product of all prime numbers.

\begin{align*}\begin{array}{rcl} && \qquad \qquad \ 144\\ && \qquad\qquad \ \big / \ \ \big\backslash \\ && \qquad\quad \ 12 \times 12 \\ && \quad \qquad \big / \ \big\backslash \quad \big / \ \big\backslash \\ && \qquad \ \ 2 \times 6 \ 2 \times 6\\ && \quad \qquad \ \ \ \big / \ \big\backslash \quad \ \big / \ \big\backslash\\ && \quad \qquad \ 2 \times 3 \ \ 2 \times 3\\ \\ && 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \end{array}\end{align*}

Next, write the factors in exponential notation.

\begin{align*}144 = 2^4 \times 3^2\end{align*}

The prime factorization of 144 is \begin{align*}2^4 \times 3^2\end{align*}.

### Review

Write the prime factorization of each number using exponential notation.

1. 56
2. 14
3. 121
4. 84
5. 50
6. 64
7. 72
8. 16
9. 24
10. 300
11. 128
12. 312
13. 525
14. 169
15. 213

To see the Review answers, open this PDF file and look for section 5.4.

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### Vocabulary Language: English

TermDefinition
Composite A number that has more than two factors.
Divisibility Rules A list of rules which help you to determine if a number is evenly divisible by another number.
Factor Tree A factor tree is a diagram for organizing factors and prime factors.
Factors Factors are numbers or values multiplied to equal a product.
Prime A prime number is a number that has exactly two factors, one and itself.
Prime Factorization The prime factorization of a number is the deconstruction of a number into the product of its primes.