2.5 Prime Factorization and the Least Common Multiple

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.

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 $36.$ We chose $12$ and $3,$ but the same result would have been the same if we had started with $2$ and $18,4$ and $9,\text{or}6\text{and}6.$

Example 2.48

Find the prime factorization of $48$ using the factor tree method.

Solution

We can start our tree using any factor pair of 48. Let's use 2 and 24. We circle the 2 because it is prime and so that branch is complete.
Now we will factor 24. Let's use 4 and 6.
Neither factor is prime, so we do not circle either. We factor the 4, using 2 and 2. We factor 6, using 2 and 3. We circle the 2s and the 3 since they are prime. Now all of the branches end in a prime.
Write the product of the circled numbers.	$2\cdot 2\cdot 2\cdot 2\cdot 3$
Write in exponential form.	$2^4\cdot 3$

Check this on your own by multiplying all the factors together. The result should be $48.$

Example 2.49

Find the prime factorization of 84 using the factor tree method.

Solution

We start with the factor pair 4 and 21. Neither factor is prime so we factor them further.
Now the factors are all prime, so we circle them.
Then we write 84 as the product of all circled primes.	$2\cdot 2\cdot 3\cdot 7$ $2^2\cdot 3\cdot 7$

Draw a factor tree of $84.$

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 $9$ 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.

Example 2.50

Find the prime factorization of $120$ using the ladder method.

Solution

Divide the number by the smallest prime, which is 2.
Continue dividing by 2 until it no longer divides evenly.
Divide by the next prime, 3.
The quotient, 5, is prime, so the ladder is complete. Write the prime factorization of 120.	$2\cdot 2\cdot 2\cdot 3\cdot 5$ $2^3\cdot 3\cdot 5$

Check this yourself by multiplying the factors. The result should be $120.$

Example 2.51

Find the prime factorization of $48$ using the ladder method.

Solution

Divide the number by the smallest prime, 2.
Continue dividing by 2 until it no longer divides evenly.
The quotient, 3, is prime, so the ladder is complete. Write the prime factorization of 48.	$2\cdot 2\cdot 2\cdot 2\cdot 3$ $2^4\cdot 3$

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 $50.$

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 $12$ 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 $12$ and the prime factors of $18$ are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that $36$ is the least common multiple.

Example 2.53

Find the LCM of $15$ and $18$ using the prime factors method.

Solution

Write each number as a product of primes.
Write each number as a product of primes, matching primes vertically when possible.
Bring down the primes in each column.
Multiply the factors to get the LCM.	$\text{LCM}=2\cdot 3\cdot 3\cdot 5$ The LCM of 15 and 18 is 90.

Example 2.54

Find the LCM of $50$ and $100$ using the prime factors method.

Solution

Write the prime factorization of each number.
Write each number as a product of primes, matching primes vertically when possible.
Bring down the primes in each column.
Multiply the factors to get the LCM.	$\text{LCM}=2\cdot 2\cdot 5\cdot 5$ The LCM of 50 and 100 is 100.