# 2.5 Prime factorization and the least common multiple

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By the end of this section, you will be able to:
• Find the prime factorization of a composite number
• Find the least common multiple (LCM) of two numbers

Before you get started, take this readiness quiz.

1. Is $810$ divisible by $2,3,5,6,\text{or}\phantom{\rule{0.2em}{0ex}}10?$
If you missed this problem, review Find Multiples and Factors .
2. Is $127$ prime or composite?
If you missed this problem, review Find Multiples and Factors .
3. Write $2\cdot 2\cdot 2\cdot 2$ in exponential notation.
If you missed this problem, review Use the Language of Algebra .

## Find the prime factorization of a composite number

In the previous section, we found the factors of a number. Prime numbers have only two factors, the number $1$ and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization    of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.

## Prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better sense of prime numbers.

You may want to refer to the following list of prime numbers less than $50$ as you work through this section.

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47$

## Prime factorization using the factor tree method

One way to find the prime factorization of a number is to make a factor tree . We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.

If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.

We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.

For example, let’s find the prime factorization of $36.$ We can start with any factor pair such as $3$ and $12.$ We write $3$ and $12$ below $36$ with branches connecting them.

The factor $3$ is prime, so we circle it. The factor $12$ is composite, so we need to find its factors. Let’s use $3$ and $4.$ We write these factors on the tree under the $12.$

The factor $3$ is prime, so we circle it. The factor $4$ is composite, and it factors into $2·2.$ We write these factors under the $4.$ Since $2$ is prime, we circle both $2\text{s}.$

The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.

$2\cdot 2\cdot 3\cdot 3$

In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.

$\begin{array}{c}2\cdot 2\cdot 3\cdot 3\\ \\ {2}^{2}\cdot {3}^{2}\end{array}$

Note that we could have started our factor tree with any factor pair of $36.$ We chose $12$ and $3,$ but the same result would have been the same if we had started with $2$ and $18,4$ and $9,\text{or}\phantom{\rule{0.2em}{0ex}}6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}6.$

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