Least Common Multiple on the GMAT

Last Updated on May 16, 2023

If you’re preparing for the GMAT, chances are you’re looking for some helpful tips on how to ace the math section. While we have written several articles on improving your GMAT quant score, this article will provide some helpful tips specifically on dealing with least common multiple (LCM) problems on the GMAT.

Here are the topics we’ll cover:

• The LCM of Two Numbers Is the Smallest Number That Is a Multiple of Both Numbers
• The Listing Method of Determining the LCM
• Prime Factorization
  • Prime Factor Tree Example 1
  • Prime Factor Tree Example 2
• Using the Prime Factorization Method to Determine the LCM
  • Least Common Multiple Example 1
  • Least Common Multiple Example 2
• Determining the LCM of Three or More Numbers
  • Least Common Multiple Example 3
  • Least Common Multiple Example 4
• Using the LCM to Determine the Least Common Denominator of a Group of Fractions
  • Least Common Multiple Example 5
• Two LCM Shortcuts
• In Summary
• Frequently Asked Questions (FAQ)
  • What Is the Lowest Common Multiple?
  • What Is the Fastest Way to Find the Lowest Common Multiple?
• What’s Next?

Let’s begin by defining the least common multiple.

The LCM of Two Numbers Is the Smallest Number That Is a Multiple of Both Numbers

Before discussing how to find the least common multiple and least common multiple strategies, let’s discuss the definition of a multiple of a number.

A multiple is a number that can be divided by another number without a remainder. For example, some multiples of 7 are 14, 21, and 28, as the remainder is zero when any of those numbers is divided by 7.

KEY FACT:

A multiple is a number that can be divided by another number with no remainder.

The least common multiple, or LCM, of two numbers is the smallest number that is a multiple of both numbers. Below are some examples of least common multiples:

• The LCM of 4 and 6 is 12 because 12 is the smallest whole number that is a multiple of both 4 and 6.
• The LCM of 3 and 2 is 6 because 6 is the smallest whole number that is a multiple of both 3 and 2.
• The LCM of 5 and 7 is 35 because 35 is the smallest whole number that is a multiple of both 5 and 7.

We should also remember that the LCM is relevant even when we have more than two numbers, but we will discuss that concept in a later section.

KEY FACT:

The least common multiple of two numbers is the smallest number that is a multiple of both numbers.

Let’s now discuss how to find the LCM of two numbers. We will discuss two methods, starting with the listing method.

The Listing Method of Determining the LCM

You may be wondering how we determined the least common multiple of the examples above. One way is simply by listing out the multiples of the two numbers until finding, yes, the least of those multiples that are common to both lists! Let’s practice this method.

What is the least common multiple of 4 and 6?

Let’s list the multiples of 4 and 6:

Multiples of 4: 4, 8, 12, 16 …

Multiples of 6: 6, 12, 18 …

The least common multiple of 4 and 6 is 12.

What is the least common multiple of 3 and 2?

Let’s list the multiples of 3 and 2:

Multiples of 2: 2, 4, 6, 8 …

Multiples of 3: 3, 6, 9 …

The least common multiple of 3 and 2 is 6.

What is the least common multiple of 5 and 7?

Let’s list the multiples of 5 and 7:

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40 …

Multiples of 7: 7, 14, 21, 28, 35, 42 …

The least common multiple of 5 and 7 is 35.

Next, let’s discuss the prime factorization method of determining the least common multiple. However, before we do, let’s discuss prime factorization in general.

Prime Factorization

We need to know how to break numbers into their prime factors to use the prime factorization method of determining the LCM. To do so, we also need to know the definition of a prime number. A prime number is an integer greater than 1 that has only itself and 1 as factors. For example, 2 is a prime number because its only factors are 2 and 1. Also, 5 is a prime number because its only factors are 5 and 1.

KEY FACT:

A prime number is an integer greater than 1 that has only itself and 1 as factors.

A factor tree is a tool that can quickly help identify a number’s prime factors. Let’s use 42 as an example to show how this tree works.

Prime Factor Tree Example 1:

First, we divide 42 by any of its factors. In this case, we will start with 7. Since 42/7 = 6, we now have two branches of our tree, pointing to 7 and 6. Since 7 is a prime number, we can leave it alone and further factor 6. Since 6/3 = 2, we finally have two more branches, pointing to 3 and 2, and since both of those numbers are prime, we can stop.

Thus, the prime factorization of 42 is 2 x 3 x 7. The factor tree is illustrated below.

Prime Factor Tree Example 2:

Let’s next prime factorize 72. First, we divide 72 by any of its factors. In this case, we can start with 9. Since 72/9 = 8, we now have two branches of our tree, pointing to 9 and 8. Next, we can further factor 9. Since 9/3 = 3, we have two more branches, pointing to 3 and 3, and since both of those numbers are prime, we can stop.

Next, we can go back to 8, and factor it further. Since 8/4 = 2, we now have two branches of our tree, pointing to 4 and 2. Since 2 is a prime number, we can leave it alone and further factor 4. Since 4/2 = 2, we finally have two more branches, pointing to 2 and 2, and since both of those numbers are prime, we can stop.

Thus, the prime factorization of 72 is 2 x 2 x 2 x 3 x 3. This can also be written as 2^3 x 3^2. The factor tree is illustrated below.

Now, let’s discuss using the prime factorization method to determine the LCM.

Using the Prime Factorization Method to Determine the LCM

We can use a second method to determine the least common multiple of a set of numbers: the prime factorization method. This method involves the following 3 steps:

Step 1:

Prime factorize each integer and put those prime factors in exponential form.

Step 2:

Pick out the largest versions of each of the prime factors. For example, imagine that you prime factorize the following two numbers:

60 = 2^2 x 3^1 x 5^1

36 = 2^2 x 3^2

The largest version of 2 is 2^2, the largest version of 3 is 3^2, and the largest version of 5 is 5^1.

Step 3:

Multiply together the prime factors selected in step 2. For example, if we were determining the LCM from the prime numbers above, we would end up with 2^2 x 3^2 x 5^1 = 4 x 9 x 5 = 180. Thus, we see that the LCM of 60 and 36 is 180.

Let’s practice with a couple of examples in which we have to determine the least common multiple of two numbers.

Least Common Multiple Example 1

Let’s practice one more.

Least Common Multiple Example 2

Determining the LCM of Three or More Numbers

So far in this article, we have discussed how to determine the LCM of just two numbers. However, we might also need to determine the LCM of three or more numbers. The good news is that we can follow the same process we’ve already learned to determine the LCM of three or more numbers. That said, when solving these types of questions, using the prime factorization method is generally advised.

TTP PRO TIP:

When determining the LCM of three or more numbers, use the prime factorization method.

Let’s practice with a couple of examples.

Least Common Multiple Example 3

Least Common Multiple Example 4

Next, let’s discuss the relationship between the least common multiple and the least common denominator.

Using the LCM to Determine the Least Common Denominator of a Group of Fractions

If you think back to when you first started learning arithmetic, you may remember learning how to add and subtract fractions. You may recall that you needed to determine the least common denominator (LCD) to do so, right? Well, you may have never considered this, but the least common denominator (LCD) is the least common multiple of your denominators! So, if you are ever struggling to come up with the least common denominator, you can use any of the methods of determining LCM that we have presented in this article.

TTP PRO TIP:

When determining the least common denominator (LCD) of a group of fractions, you can use the techniques for finding the least common multiple.

Let’s practice with an example.

Least Common Multiple Example 5

Lastly, let’s discuss a couple of least common multiple shortcuts.

Two LCM Shortcuts

We have discussed two methods for determining the least common multiple of a group of numbers. While those methods will work 100 percent of the time, we can also use a few shortcuts, depending on the types of numbers given.

The first shortcut is useful when the given numbers do not share the same prime factors. In this case, the LCM is the product of the two numbers. For example:

• The LCM of 25 and 4 is 25 x 4 = 100. Since 25 has only prime factors of 5, and since 4 has only prime factors of 2, we see that they have no prime factors in common. Thus, we know that the LCM is the product of 25 and 4.
• The LCM of 16 and 9 is 16 x 9 = 144. Since 16 has only prime factors of 2, and since 9 has only prime factors of 3, we see that they have no prime factors in common. Thus, we know that the LCM is the product of 9 and 16.

TTP PRO TIP:

When two or more numbers do not share any of the same prime factors, the LCM is the product of those numbers.

The second shortcut is useful if, when we’re given two numbers, the larger number is a multiple of the smaller number. In that case, the LCM is the larger number. For example:

• The LCM of 48 and 8 is 48 because 48 is a multiple of 8.
• The LCM of 15 and 75 is 75 because 75 is a multiple of 15.

This shortcut works for more than two numbers as well.

TTP PRO TIP:

When given two or more numbers, if the largest number is a multiple of each of the smaller numbers, then the LCM is the largest number.

In Summary

The Least Common Multiple (LCM) is an important concept in GMAT quant. What the LCM is, how to determine it, and how to use it are all important concepts tested on the GMAT.

Here are the key concepts we’ve discussed:

• A multiple is a number that can be divided by another number with no remainder.
• The least common multiple (LCM) of two or more numbers is a multiple of all the numbers. It is the smallest of the common multiples that are shared by all of those numbers.
• The listing method can be used to determine the LCM of a set of numbers. First, list multiples of each of the numbers. Then, find multiples that appear in all the lists. The LCM is the smallest of these common multiples.
• The more sophisticated method of finding the LCM is the prime factorization method, in which each number is prime factored. Then, the prime factors with the largest exponent (for each prime factor) are multiplied. The resulting product is the LCM.
• Either the listing method or the prime factorization method can be used for any group of numbers for which you need to find the LCM. However, the prime factorization method is preferred because it is generally more efficient.
• The LCM techniques are used to find the Least Common Denominator (LCD) for fractions with unequal denominators. The LCD is the same as the LCM of fractions’ denominators.
• Two shortcuts exist for finding the LCM (or LCD) of certain sets of numbers:
  • If two or more numbers do not share any prime factors, the LCM is the product of those numbers.
  • If the largest number of a set of two or more numbers is a multiple of the smaller number(s), then the LCM of the numbers is that largest number.

Frequently Asked Questions (FAQ)

What Is the Lowest Common Multiple?

The least (or lowest) common multiple of two numbers is the smallest number that is a multiple of both numbers.

What Is the Fastest Way to Find the Lowest Common Multiple?

There is no one fastest way to find the LCM, as the fastest way depends on the numbers we’re using.

In this article, we have presented both the list method and the prime factorization method, either of which can be used to find the LCM of any set of numbers. However, in general, the prime factorization method is more efficient.

What’s Next?

The LCM is just one of hundreds of math topics you’ll encounter in the quant section of the GMAT. You might feel overwhelmed at times, but remember, while mastering these topics requires time and effort, GMAT quant can be conquered.

Need more help mastering GMAT quant topics? Try the TTP GMAT course for just $1!

Good luck!

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