Finding the Least Common Multiple using the List Method
The least common multiple, also known as the LCM, of two numbers is the smallest number that is divisible by the two given numbers. The assumption here is that the numbers involved are positive whole numbers or positive integers
But first, we must ask ourselves. What is a multiple of a number?
Suppose we have two positive whole numbers [latex]n[/latex] and [latex]m[/latex]. The number [latex]m[/latex] is a multiple of the number [latex]n[/latex] if [latex]n[/latex] can evenly divide [latex]m[/latex]. That means when [latex]m[/latex] is divided by [latex]n[/latex], the result has a remainder of zero.
For instance, [latex]20[/latex] is a multiple of [latex]10[/latex] since [latex]20[/latex] divided by [latex]10[/latex] equals [latex]2[/latex] and more importantly, it has NO remainder.
Another way of looking at it is that a multiple of a number is the product of the given number and a natural or counting number.
For example, the number [latex]54[/latex] is a multiple of [latex]6[/latex] because [latex]54 = 6 \times 9[/latex]. Notice that the number [latex]6[/latex] is being multiplied to a counting number which is [latex]9[/latex].
This next concept may sound trivial but it is very important. A number itself is its own multiple. It is obvious to see that [latex]5[/latex] is a multiple of [latex]5[/latex] because [latex]5[/latex] divided by [latex]5[/latex] is [latex]1[/latex] and without a remainder.
Or, [latex]5[/latex] is a multiple of itself since [latex]5 = 5 \times 1[/latex] where the number [latex]5[/latex] is being multiplied to the counting number [latex]1[/latex].
Now it is time for us to learn how to list the multiples of a given number. Bear in mind that for any given positive whole number, it has an infinite number of multiples.
Let’s take a look at the multiples of [latex]7[/latex].
Here’s the trick! To find the multiples of [latex]7[/latex], start by writing the number itself then we skip count by [latex]7[/latex].
Therefore, the multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, …
The “…” symbol, also known as ellipses, implies that the sequence goes on without end but follows a certain pattern.
Another way of generating the multiples of a number is to make use of the set of natural numbers. Remember, the set of the natural numbers (also known as the set of counting numbers) contains the elements 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …
We can also express the counting numbers as a set.
We will use the set of counting numbers as the multipliers to a given number to generate its multiples. Because a number has infinite multiples, we will need to specify how many multiples we want to list. For the sake of this lesson, let’s agree to write or list the first eight (8) multiples of a number.
Below is a list of the first eight multiples of [latex]6[/latex]. Notice that to find them, we will multiply [latex]6[/latex] by the first eight elements of the set of counting numbers which are 1, 2, 3, 4, 5, 6, 7, and 8. The products become the first eight multiples of [latex]6[/latex].
Let’s go over more examples on finding the multiples of a number. The more examples that you see, the more comfortable you become with the concept.
◉ First five multiples of 3 ☞ 3, 6, 9, 12, 15
◉ First seven multiples of 10 ☞ 10, 20, 30, 40, 50, 60, 70
◉ First eight multiples of 9 ☞ 9, 18, 27, 36, 45, 54, 63, 72
◉ First ten multiples of 13 ☞ 13, 26, 39, 52, 65, 78, 91, 104, 117, 130
Examples of Finding the Least Common Multiple
1) Find the least common multiple of [latex]3[/latex] and [latex]7[/latex].
The skills that we have learned how to find the multiples of a number will come into play here. The only difference is that we will find the multiples of two numbers and we will list them side by side.
It is up to us how many multiples that we decide to write. Sometimes we will have the need to extend it because we cannot find the first common multiple just yet. The first number that shows up on the list that is common to both becomes the least common multiple or LCM of the given two numbers.
So let’s write down the first ten multiples of [latex]3[/latex] and [latex]7[/latex] and see if we could find the first match. If we have done it correctly, the LCM of 3 and 7 is 21.
Remember that the key here is to find the common multiple that has the least in value.
It is very possible to have more than one common multiples. But when it comes to finding the least common multiple, we are definitely interested in finding the smallest common multiple. Please check the diagram below. I hope it makes a lot of sense!
2) Find the least common multiple of [latex]8[/latex] and [latex]12[/latex].
I hope you already get the hang of it. Let’s work this out step by step.
- List the first ten multiples of 8 and 12.
- Identify the multiples that are common to both lists. As you can see in the illustration below, the common multiples of 8 and 12 are 24, 48 and 72. Just to clarify, these are the common multiples of 8 and 12 for their first ten multiples.
- The common multiple which has the smallest in value is the least common multiple (LCM) of the given two numbers which are 8 and 12. In this case, the LCM of 8 and 12 is 24.
3) What is the LCM of [latex]14[/latex] and [latex]20[/latex] ?
As you can see, the problems in finding the LCM of two numbers can become more challenging as the numbers increase. Since you already know the procedure, the entire process should be manageable to you.
The common mistake that most of my students commit is when they become careless in writing down the first few multiples of the numbers. So don’t fall into the trap of being complacent. Apply the things that you’ve learned and execute it with purpose.
I suggest that you work this out first on paper before clicking the button to reveal the solution for each step. Good luck!
- Step 1: Write the first twelve multiples of 14 and 20.
Step # 1
- Step 2: Mark the common multiples of 14 and 20.
Step # 2
- Step 3: Identify the least common multiple (LCM) of 14 and 20.
Step # 3
4) What is the LCM of [latex]11[/latex] and [latex]23[/latex] ?
This is not a trick question. I would say that this is a perfectly fair question to ask in a test. This type of problem regarding LCM is something we math teachers always like to throw into the mix to test the students’ understanding of the topic.
So what should you do? As always, for every math problem, try to step back to look at the problem in a bigger picture. Just don’t get into the habit of immediately solving the problem without having a good plan. Because some problems may look daunting at first which can cause math anxiety, when in fact it is very easy as long as you know what you are dealing with.
First, what can you say about the two numbers [latex]11[/latex] and [latex]23[/latex]? Are they somehow special?
The answer is yes! The numbers [latex]11[/latex] and [latex]23[/latex] are both prime numbers. That means [latex]11[/latex] is only divisible by [latex]1[/latex] and itself. The same reasoning goes with [latex]23[/latex] that it is only divisible by [latex]1[/latex] and itself.
The rule states that if [latex]a[/latex] and [latex]b[/latex] are two distinct prime numbers, their least common multiple (LCM) is just their product, that is, [latex]a \times b[/latex].
Since we have already established that [latex]11[/latex] and [latex]23[/latex] are prime numbers, their LCM is simply their product which [latex]11 \times 23 = 253[/latex]. We can also write our final answer as LCM (11, 23) = 253.
Now, suppose you don’t know this rule. You have no choice but to list enough multiples for each number such that you hit the first match. Your usual solution may look something like below. Imagine the possibility of incorrectly writing the multiples of 11 and 23 and therefore not getting the correct LCM. Yes, it can be really messy!
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