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GMAT Algebra: How to Find Least Common Multiples

Finding the Least Common Multiple (LCM) of two numbers is a necessary and useful skill that will enable you to solve a number of different types of GMAT problem solving and GMAT data sufficiency questions. First let’s take a look at the technical aspect of how you actually find the LCM of two numbers, and then we’ll see how this concept is applied on the GMAT.

Finding the Least Common Multiple (LCM)

The foundational principle you must know to find the Least Common Multiple of two numbers is Prime Factorization. Just as we used prime factorization as the starting point for finding the Greatest Common Factor (GCF) of two numbers, it’s also the starting point for finding the LCM of those numbers. If you’re not familiar with the technique and usage of Prime Factorization on the GMAT, start by reviewing our article titled “Effectively Using Prime Factorization on the GMAT.”

Now for some definitions.

Least Common Multiple (LCM): The LCM of two or more numbers is the smallest number that is divisible by those original integers.

Rule (How to find the LCM): The LCM of two or more numbers is equal to the product of each factor by the maximum number of times it appears in the prime factorization of those numbers.

What does this mean in plain English? Let’s take a look at this example:

Q: What is the Least Common Multiple of 60 and 100?

Step 1: Find the prime factorizations of 60 and 100. In the prime factorization diagram above you see that:
60 = 2 * 2 * 3 * 5
– 100 = 2 * 2 * 5 * 5

Step 2: Determine the most times each prime factor appears in either factorization.

2 appears twice in 60 (and 100 for that matter!)

3 appears once in 60

5 appears twice in 100

Step 3: To find the LCM of 60 and 100, apply the rule noted above.

The LCM is “the product of each factor by the maximum number of times it appears in the prime factorization of either number.” In this case, we’ll use 2 twice, 3 once, and 5 twice in the final product. Thus, the LCM of 60 and 100 = 2 * 2 * 3 * 5 * 5 = 300.

That’s it! It’s pretty easy once you get the hang of it.

Application: LCMs on the GMAT

Now that you know how to find the LCM of two numbers, how is this useful on the GMAT? How do we apply what we just learned? Let’s take a look at an application question from the world of GMAT Problem Solving.

Sample GMAT LCM Question (excerpted from Brandon Royal’s Game Plan for the GMAT) The Royal Hawaiian Hotel decorates its Rainbow Christmas Tree with non-flashing white lights and a series of colored flashing lights — red, blue, green, orange, and yellow. The red lights turn red every 20 seconds, the blue lights turn blue every 30 seconds, the green lights turn green every 45 seconds, the orange lights turn orange every 60 seconds, and yellow lights turn yellow every 1 minute and 20 seconds. The manager plugs the tree in for the first time on December 1st precisely at midnight and all lights begin their cycle at exactly the same time. If the five colored lights flash simultaneously at midnight, what is the next time all five colored lights will all flash together at the exact same time?

(A) 0:03 AM
(B) 0:04 AM
(C) 0:06 AM
(D) 0:12 AM
(E) 0:24 AM

The first question you may have when looking at this problem is, “How in the world would I know that finding the LCM is the best way to go about solving a question like this?!”

That’s a legitimate question. Let’s tackle that before actually solving the problem itself.

There are two things you could recognize that might possibly trigger the realization that prime factorization and finding the LCM is a good place to start on a problem like this. First, it might dawn on you if you start by trying to solve this problem via the Trial-and-Error Method. By the way, that’s a perfectly acceptable strategy for solving this problem, and it can definitely get you a right answer.

Let’s jump inside your brain for a moment and think about how that trial-and-error thought process might go: “Okay, the Red lights flash every 20 second, so it’ll complete 3 cycles in 1 minute. Do all of the other lights somehow line up again with one minute? Well, the Blue lights flash every 30 seconds, so it’ll complete 2 cycles in 1 minute. The Blue and Red lights therefore line up at minute increments. How about the Green lights? Uh-oh. They flash every 45 seconds, so the Green lights won’t line up with the Red and Blue lights until….hmmm, let’s see….how many cycles will it take to line up with an even number of minutes? Ahhh, got it! After 4 cycles, the Green lights flash again at the 3-minute mark….”

…and so on. You could definitely come to the right answer by going through all 5 light sequences that way.

BUT, hopefully at some point it would dawn on you that what you’re really doing is trying to find the LCM of all of those individual flash times! In other words, How many seconds are divisible by ALL five of those individual increments? That’s the definition of LCM, my friends!

In fact, that’s the second way you could have recognized that this is a LCM question. The key words in the question are “line up.” To find the point at which a series of objects “line up,” find the Least Common Multiple of the numbers involved.

So, let’s use the Prime Factor / LCM Method to solve this problem and see how much easier it is than trial-and-error. You might organize it on your scratch paper something like this:

 COLOR FLASH TIME PRIME FACTORS Red 20 seconds 2 x 2 x 5 = 20 Blue 30 seconds 2 x 3 x 5 = 30 Green 45 seconds 3 x 3 x 5 = 45 Orange 60 seconds 2 x 2 x 3 x 5 = 60 Yellow 80 seconds 2 x 2 x 2 x 2 x 5 = 80

Therefore, the LCM of all 5 colors is 2 x 2 x 2 x 2 x 3 x 3 x 5 = 720 seconds. 720 seconds = 12 minutes, so the correct answer is D.

Great job if you got Answer D on your own, by the way!

Make sure to add Prime Factorization and LCMs to your toolbox of GMAT math skills, and keep your antennae up on test day for questions that lend themselves to finding the Least Common Multiple of two or more numbers.

Oh yeah, and the best part about mastering a sample GMAT question like this? Now you’ll know how to coordinate your Christmas lights next season and impress all of your neighbors! Who says the GMAT doesn’t have any practical real-world applications?!