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## Prime Factorization

#### Introduction

Prime Factorization is expressing a number as a product of its prime factors. For example, the prime factorization of the number 18 can be written as $2 ast 3 ast 3$ or $3^{2} ast 2$.

#### Terms

Factor - Factors are numbers multiplied together to get another number. Because 3 * 4 = 12, both 3 and 4 are factors of 12.
Prime - A prime number is a number that only has 1 and itself as factors.
Prime Factor - A prime factor is a factor that is also a prime number.

## Lesson

To find the prime factorization for a given postive integer, we use what is called a factor tree, which looks like this for the number 12:

The prime factors are represented by the circled values, 2, 2, and 3. So for 12, the prime factorization is $2 ast 2 ast 3$, or $2^{2} ast 3$.

If a number’s only factors are only 1 and itself, then that number is a prime number and no further factorization is possible.

## Examples

#### Prime Factorization of 12

$12$
$12 = 2 * 6$
$6 = 2 * 3$
The prime factorization of 12 can be written as:
$2^{2} * 3^{1}$

#### Prime Factorization of 72

$72$
$72 = 2 * 36$
$36 = 2 * 18$
$18 = 2 * 9$
$9 = 3 * 3$
The prime factorization of 72 can be written as:
$2^{3} * 3^{2}$

#### Prime Factorization of 61

$61$
The factors of $61$ are $1\text{ and }61$. Therefore $61$ is a prime number.