# The Language of Algebra

## Prime Factorization and the Least Common Multiple


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

### 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  We can start with any factor pair such as  and  We write  and  below  with branches connecting them.
The factor  is prime, so we circle it. The factor  is composite, so we need to find its factors. Let’s use  and  We write these factors on the tree under the 
The factor  is prime, so we circle it. The factor  is composite, and it factors into  We write these factors under the  Since  is prime, we circle both 
The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.
In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.
Note that we could have started our factor tree with any factor pair of  We chose  and  but the same result would have been the same if we had started with  and  and 

### Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.
To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for  we divide  by  the smallest prime factor of 
To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.
Then we divide by the next prime; so we divide  by 
We continue dividing up the ladder in this way until the quotient is prime. Since the quotient,  is prime, we stop here.
Do you see why the ladder method is sometimes called stacked division?
The prime factorization is the product of all the primes on the sides and top of the ladder.
Notice that the result is the same as we obtained with the factor tree method.

### Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

### Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of  and  We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.
We see that  and  appear in both lists. They are common multiples of  and  We would find more common multiples if we continued the list of multiples for each.
The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of  and  is 

### Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of  and 
We start by finding the prime factorization of each number.
Then we write each number as a product of primes, matching primes vertically when possible.
Now we bring down the primes in each column. The LCM is the product of these factors.
Notice that the prime factors of  and the prime factors of  are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that  is the least common multiple.

### Key Concepts

1. Find the prime factorization of a composite number using the tree method.

2. Find the prime factorization of a composite number using the ladder method.

3. Find the LCM by listing multiples.

4. Find the LCM using the prime factors method.


### Section Exercises


### Practice Makes Perfect

Find the Prime Factorization of a Composite Number 
In the following exercises, find the prime factorization of each number using the factor tree method.
In the following exercises, find the prime factorization of each number using the ladder method.
In the following exercises, find the prime factorization of each number using any method.
Find the Least Common Multiple (LCM) of Two Numbers 
In the following exercises, find the least common multiple (LCM) by listing multiples.
In the following exercises, find the least common multiple (LCM) by using the prime factors method.
In the following exercises, find the least common multiple (LCM) using any method.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


### Chapter Review Exercises


### Use the Language of Algebra

Use Variables and Algebraic Symbols 
In the following exercises, translate from algebra to English.
Identify Expressions and Equations 
In the following exercises, determine if each is an expression or equation.
Simplify Expressions with Exponents 
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
In the following exercises, simplify each expression.
Simplify Expressions Using the Order of Operations 
In the following exercises, simplify.

### Evaluate, Simplify, and Translate Expressions

Evaluate an Expression 
In the following exercises, evaluate the following expressions.
Identify Terms, Coefficients and Like Terms 
In the following exercises, identify the terms in each expression.
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
Simplify Expressions by Combining Like Terms 
In the following exercises, simplify the following expressions by combining like terms.
Translate English Phrases to Algebraic Expressions 
In the following exercises, translate the following phrases into algebraic expressions.

### Solve Equations Using the Subtraction and Addition Properties of Equality

Determine Whether a Number is a Solution of an Equation 
In the following exercises, determine whether each number is a solution to the equation.
Model the Subtraction Property of Equality 
In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.
Solve Equations using the Subtraction Property of Equality 
In the following exercises, solve each equation using the subtraction property of equality.
Solve Equations using the Addition Property of Equality 
In the following exercises, solve each equation using the addition property of equality.
Translate English Sentences to Algebraic Equations 
In the following exercises, translate each English sentence into an algebraic equation.
Translate to an Equation and Solve 
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Mixed Practice 
In the following exercises, solve each equation.

### Find Multiples and Factors

Identify Multiples of Numbers 
In the following exercises, list all the multiples less than  for each of the following.
Use Common Divisibility Tests
In the following exercises, using the divisibility tests, determine whether each number is divisible by 
Find All the Factors of a Number 
In the following exercises, find all the factors of each number.
Identify Prime and Composite Numbers 
In the following exercises, identify each number as prime or composite.

### Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number 
In the following exercises, find the prime factorization of each number.
Find the Least Common Multiple of Two Numbers 
In the following exercises, find the least common multiple of each pair of numbers.

### Everyday Math


### Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.
In the following exercises, identify each as an expression or equation.
In the following exercises, simplify, using the order of operations.
In the following exercises, evaluate each expression.
In the following exercises, translate each phrase into an algebraic expression.
In the following exercises, solve each equation.
In the following exercises, translate each English sentence into an algebraic equation and then solve it.