By the end of this section, you will be able to:
In the previous section, we found the factors of a number. Prime numbers have only two factors, the number 11 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 50 as you work through this section.
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 36.36. We can start with any factor pair such as 33 and 12.12. We write 33 and 1212 below 3636 with branches connecting them.
The factor 33 is prime, so we circle it. The factor 1212 is composite, so we need to find its factors. Let’s use 33 and 4.4. We write these factors on the tree under the 12.
The factor 33 is prime, so we circle it. The factor 44 is composite, and it factors into 2⋅2.2·2. We write these factors under the 4.4. Since 22 is prime, we circle both 2s.
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 36.36. We chose 1212 and 3,3, but the same result would have been the same if we had started with 22 and 18,4 and 9,or6 and 6
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 36, we divide 36 by 2, the smallest prime factor of 36.
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 99 by 3.
We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, 3, 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 10 and 25. 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 50 and 100 appear in both lists. They are common multiples of 10 and 25. 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 10 and 25 is 50.
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 12 and 18.
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 1212 and the prime factors of 1818 are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that 3636 is the least common multiple.
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 least common multiple (LCM) by listing multiples.
Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)314.
Grocery shopping Paper plates are sold in packages of 1212 and party cups come in packs of 8.8. What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)
ⓐ 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?