Using Prime Factorization to Find LCM

From my previous lesson, I went over the steps on how to find the LCM of two positive integers using the List Method. This time I will focus on the method where Prime Factorization is used to find the LCM. 

I highly recommend that you review the procedure on how to perform Prime Factorization because this skill will play a major role in this lesson. Check out the following link: Integer Prime Factorization.


Steps on How to Find the LCM using Prime Factorization

Step 1: Perform the prime factorization of each number then write it in exponential form. Align the common prime factor base whenever possible.

Step 2: For the numbers with a common prime factor base, select the prime number that has the highest power. The prime factor with the highest power implies that it occurs the most in the entire list.

Step 3: If a distinct prime factor has NO matching prime factor base in the list, immediately include this factor with its exponent in the collection of numbers that you will multiply later.

Note: Steps #2 and #3 ensure that all distinct prime factors in the ENTIRE list are represented without duplicates.

Step 4: To determine the Least Common Multiple (LCM), multiply all the numbers that you have collected or gathered from steps #2 and #3.


Examples of Determining the Least Common Multiple (LCM) using Prime Factorization

Example 1: What is the LCM of 12 and 90 ?

First, write the prime factorization of each number in exponential form. Make sure to align the numbers that have a common base. If a number does not have a common base, then write it in a way that there’s nothing above or below it to indicate that it is unique.

Observe that {2^2} and 2 are vertically aligned to signify they have a common base of 2. In the same manner, 3 and {3^2} are aligned because they have a common base of 3. However, 5 has a unique base so we write it by itself on its own “column”.

twelve is equal to 2 raised to the second power times three

ninety is equal to two times 3 raised to the second power times five

For emphasis, I labelled the prime factorizations of 12 and 90. I hope it makes a lot more sense now.

Now, it is time to choose the numbers that we are going to multiply to get the LCM.

  • First, between {2^2} and 2, we will select {2^2} because it has a higher power than 2.
  • Secondly, between 3 and {3^2}, we choose {3^2} for the same reason.
  • Finally, since the base 5 is by itself and we have nothing to compare it with, we will automatically select it to be included in the group.
  • Therefore, the numbers that we are going to multiply to determine the LCM of 12 and 90 are {2^2}, {3^2} and {5}.
  • Multiplying the numbers we have selected, we get:
LCM(12,90) = 4 times 9 times 5 = 180

Example 2: What is the LCM of 80 and 120 ?

This time the numbers are relatively larger than the previous example. However, don’t fret because as long as you stick to the procedure, the entire solving process for LCM is going to be smooth sailing. That’s the testament to the power of prime factorization. It doesn’t matter how big the integers are because they can be expressed as prime factors. 

As you may have already observed, being able to correctly prime factorize a positive integer is an indispensable skill to acquire and master. Once you have it, nobody can take it away from you. 

Begin by prime factorizing 80 and 120. Observe that the numbers with bases of 2 and 5 are respectively aligned vertically. The prime factor 3 is by itself so there’s no number above it.

Below is the same image but with labels for emphasis.

Let’s pick the numbers to multiply to determine the least common multiple.

  • Between {2^4} and {2^3}, we pick {2^4} over {2^3} because the former has a higher power than the latter since 4 > 3.
  • The number 3 is by itself so we will automatically include it in the collection of numbers that we are going to multiply.
  • The numbers 5 and 5 are exactly the same. We will just pick one copy of 5 to represent the two.
  • Therefore, the numbers {2^4}, 3, and 5 are the ones we will multiply to the LCM of
80 = 2^4 times 5

120 = 2^3 times 3 times 5
  • So the LCM for 80 and 120 is calculated as follows:
LCM(80,120)=2^4*3*5=16*3*5=240

Example 3: What is the LCM of 17 and 71 ?

Don’t rush into solving the problem because you are already familiar with the steps. It wouldn’t hurt to step back for a moment to look at the problem in a bigger picture. 

In this case, you should be able to spot immediately that both 17 and 71 are prime numbers. Remember that a prime number is only divisible by 1 and itself. Just a quick reminder, don’t forget the fact that the smallest prime number is 2 and NOT 1.

So if we are going to perform prime factorizations on a prime number, you should be convinced that the prime factorization of a prime number is just the prime number itself. Please don’t include 1 since it is not a prime number.

Here are the prime factorizations of the prime numbers 17 and 71. I didn’t align them in the same column because they don’t have a common base.

The next obvious step is to choose the numbers that we are going to multiply. Since the bases are different from each other then we must select both of them.

To find the LCM of 17 and 71, we have:

LCM of 17 and 71 is equal to 17 times 71 which is 1,207.

Example 4: What is the LCM of 126, 252 and 336 ?

Let’s kick it up a notch by finding the least common multiple (LCM) of three distinct positive integers that are in the hundreds.

I know that it can be intimidating because it feels like we are in a new territory. However, if you are already familiar with prime factorization there is nothing you should worry about. You got this!

The steps or procedure in finding the LCM of three integers is very similar to finding the LCM of two numbers which at this point you should have a mastery already.

Start by applying Prime Factorization on the positive integers 126, 252 and 336. Again, it is so important that we align the numbers that have a common base. As you can observe, we vertically aligned the numbers with a base of 2, a base of 3, and a base of 7. This simple trick gives us enormous heads up what to do next.

126 = 2 times 3^2 times 7
252 = 2^2 times 3^2 times 7
336 = 2^4 times 3 times 7

The following illustration simply emphasizes the fact that we aligned the numbers that have the same base.

The next step as you already know is to select the numbers that we are going to multiply to determine the least common multiple of the three numbers.

  • With a common base of 2, we select {2^4} since it has the highest power.
  • With a common base of 3, we select {3^2} because it has the highest power. Since there are two {3^2}, just pick one instance of it.
  • With a common base of 7, we select 7 because they are all the same.

The least common multiple (LCM) of 126, 252 and 336 is the product of the numbers that we have selected above. They are marked in orange.

LCM(125,252,336) is equal to 16 times nine times seven which is one thousand eight.

You might also be interested in:

Finding LCM using the List Method

Finding GCF using the List Method

Use Prime Factorization to Find GCF