Use List Method to Find LCM (2024)

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

Use List Method to Find LCM (1)

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.

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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].

Use List Method to Find LCM (3)

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.

Use List Method to Find LCM (4)

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].

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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 33, 6, 9, 12, 15

◉ First seven multiples of 1010, 20, 30, 40, 50, 60, 70

◉ First eight multiples of 99, 18, 27, 36, 45, 54, 63, 72

◉ First ten multiples of 1313, 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.

Use List Method to Find LCM (6)

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!

Use List Method to Find LCM (7)

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.
Use List Method to Find LCM (8)
  • 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.
Use List Method to Find LCM (9)
  • 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.
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3) What is the LCM of [latex]14[/latex] and [latex]20[/latex] ?

Use List Method to Find LCM (11)

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
Use List Method to Find LCM (12)
  • Step 2: Mark the common multiples of 14 and 20.
Step # 2
Use List Method to Find LCM (13)
  • Step 3: Identify the least common multiple (LCM) of 14 and 20.
Step # 3
Use List Method to Find LCM (14)

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.

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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!

Use List Method to Find LCM (16)

You may also be interested in these related math lessons or tutorials:

Use Prime Factorization to Find LCM

Finding GCF using the List Method

Use Prime Factorization to Find GCF

Use List Method to Find LCM (2024)

FAQs

Use List Method to Find LCM? ›

Step 1: List the first few multiples of A and B. Step 2: Mark the common multiples from the multiples of both numbers. Step 3: Select the smallest common multiple. That lowest common multiple is the LCM of the two numbers.

How to find LCM by listing method? ›

Step 1: List the first few multiples of A and B. Step 2: Mark the common multiples from the multiples of both numbers. Step 3: Select the smallest common multiple. That lowest common multiple is the LCM of the two numbers.

How to find LCM of a list in Python? ›

We use a while loop that continues until we find a number (lcm) that is divisible by all the numbers in the list. We use the all() function along with a generator expression to check if lcm is divisible by each number in the list. Once we find such an lcm, we break out of the loop and return the calculated LCM.

How do you find LCM and GCF using listing method? ›

For example, the LCM of 5 and 6 can be found by simply listing the multiples of 5 and 6, and then identifying the lowest multiple shared by both numbers. 30 is the LCM. Similarly, the GCF can be found by listing the factors of each number, and then identifying the greatest factor that is shared.

How to do listing method? ›

Listing Method

This method involves writing the members of a set as a list, separated by commas and enclosed within curly braces. For example, the four seasons are a set and could be written as {Summer, Autumn, Spring, Winter}. Note: The order of the elements in the list doesn't matter.

What is the LCM of 7 and 6 using the listing method? ›

LCM of 6 and 7 is 42. Among all common multiples of 6 and 7, the LCM of 6 and 7 is the smallest number. (6, 12, 18, 24, 30, 36, and so on) and (7, 14, 21, 28, 35, 42, 49, and so on) are the first few multiples of 6 and 7.

What is the method of finding LCM? ›

LCM by Division Method

In this method, divide the given numbers by common prime number until the remainder is a prime number or one. LCM will be the product obtained by multiplying all divisors and remaining prime numbers. Example 2: Find the LCM of 24 and 15 by the division method.

What is the LCM of 3 and 8 using listing method? ›

The LCM of 3 and 8 is 24. To find the least common multiple (LCM) of 3 and 8, we need to find the multiples of 3 and 8 (multiples of 3 = 3, 6, 9, 12 . . . . 24; multiples of 8 = 8, 16, 24, 32) and choose the smallest multiple that is exactly divisible by 3 and 8, i.e., 24.

What is the rule method of listing method? ›

The listing method is the method in which the members of the set are written as a list, separated by the comma and enclosed within the curly braces. Rule method in sets involves specifying the rule or a condition that can be used to decide whether an object can belong to the set.

What is the list method of GCF? ›

GCF by Listing out the Factors (List Method) In this method, we write down all the factors/divisors of a group of numbers. After listing down the divisors, we pick the greatest number that commonly divides the said numbers without leaving any remainder.

What is an example of the GCF listing method? ›

For example, the GCF of 14 and 35 is 7. By using the listing common factors method, the factors of 14 are 1, 2, 7, 14 and the factors of 35 are 1, 5, 7, 35. The two common factors are 1 and 7 out of which 7 is the largest. Therefore, 7 is the GCF of 14 and 35.

What is the LCM of 6 and 8 using the listing method? ›

The LCM of 6 and 8 is 24. To find the least common multiple (LCM) of 6 and 8, we need to find the multiples of 6 and 8 (multiples of 6 = 6, 12, 18, 24; multiples of 8 = 8, 16, 24, 32) and choose the smallest multiple that is exactly divisible by 6 and 8, i.e., 24.

What is the LCM of 8 and 12 using the listing method? ›

The LCM of 8 and 12 is 24. To find the least common multiple (LCM) of 8 and 12, we need to find the multiples of 8 and 12 (multiples of 8 = 8, 16, 24, 32; multiples of 12 = 12, 24, 36, 48) and choose the smallest multiple that is exactly divisible by 8 and 12, i.e., 24.

What is the LCM of 3 and 6 using listing method? ›

LCM of 3 and 6 Solved Example

LCM is the smallest number exactly divisible by 3 and 6. Multiples of 3 = 3, 6, 9, 12, 15, …. Multiples of 6 = 6, 12, 18, 24, 30, ….. Hence, the LCM of 3 and 6 is 6.

What is the LCM of 4 and 7 using listing method? ›

LCM of 4 and 7 is 28. The Least Common Multiple, or LCM, is the smallest number that can be evenly split into all of the numbers being addressed, whether they are two or more. The smallest number among all common multiples of 4 and 7 is the LCM of 4 and 7. (4, 8, 12, 16, 20, etc.)

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