Least Common Multiples (LCMs)

A common multiple of two whole numbers a and b is any number c which a and b both divide into evenly. For example, 48 is a common multiple of 6 and 12 because there are no remainders in either of the following equations:

$\frac{48}{6} = 8$

$\frac{48}{12} = 4$

The least common multiple (or LCM for short) is exactly what it sounds like. The smallest number among the common multiples for any group of numbers. There are three ways to determine the LCM for a group of numbers, and this article will introduce you to all three of them. Let's get started!

Listing multiples to find the LCM

The most straightforward way of finding the LCM between any two numbers is simply listing their multiples until one number appears on both lists. For instance, let's say you wanted to find the LCM between 9 and 12. You could create two lists as follows:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, etc.

12, 24, 36, 48, 60, 72, 84, 96, etc.

Since 36 is the smallest number appearing on both lists, you have your answer. 72 also appears on both lists, making it a common multiple of 9 and 12. However, 72 is larger than 36 so it isn't the least common multiple.

Essentially, you're counting by the numbers you're working with until you find a match. This methodology can work for any group of numbers but may become cumbersome with larger numbers. After all, nobody really counts by 247s or 532s. When the numbers get bigger, you may want a new approach.

Using the GCF to find the LCM

If you only have two numbers, you can find the LCM by dividing their product by their greatest common factor. For instance, let's say that you want to know the LCM for 18 and 30. The first step is determining what their GCF is through prime factorization:

18: $2 \times 2 \times 3$

30: $2 \times 3 \times 5$

The common factors are 2 and 3, which can be multiplied to get a GCF of 6. Now we can find the LCM by multiplying the two numbers together and dividing the product by the GCF. You can make the calculation a little more manageable by canceling a common factor. That gives us:

$18 \times \frac{30}{6} = 3 \times 6 \times \frac{30}{6} = 90$

90 is the LCM for 18 and 30. If the numbers share no common factors other than 1, the LCM is simply equal to the product of the numbers. For example, 10 and 27 shares no common factors other than 1. Therefore, the LCM would be $270 (10 \times 27)$ .

This method will help you work with larger numbers, but it doesn't always work if you have more than two numbers. Fortunately, the next methodology will.

Using prime factors to find the LCM

The prime factor method of finding the LCM begins by listing the prime factors of every number you're working with. If our numbers are 16, 25, and 60, that would look like this:

$16 : 2 \times 2 \times 2 \times 2 = 2^{4}$

$25 : 5 \times 5 = 5^{2}$

$60 : 2 \times 2 \times 3 \times 5 = 2^{2} \times 3 \times 5$

Next, we want to look at the highest power of each factor from the above list. The first list gives us four 2s, the second gives 2 5s, and the third one has a single 3. Now we want to multiply each highest powered factor. That means that:

$60 : 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 3 = 2^{2} \times 5^{2} \times 3$

It's a lot of multiplication, but you'll eventually get an LCM of 1200.

Least common multiples (LCMs) practice questions

a. What is the LCM of 7 and 8?

Prime factors of 7: 7 (it's a prime number)

Prime factors of 8: $2^{3} (2 \times 2 \times 2)$

$LCM (7, 8) = 7^{1} \times 2^{3} = 7 \times 8 = 56$

Answer: 56

b. Find the LCM of 19 and 6.

Prime factors of 19: 19 (it's a prime number)

Prime factors of 6: $2 \times 3$

$LCM (19, 6) = 19^{1} \times 2^{1} \times 3^{1} = 19 \times 2 \times 3 = 114$

Answer: 114

c. Find the LCM of 21 and 33.

Prime factors of 21: $3 \times 7$

Prime factors of 33: $3 \times 11$

$LCM (21, 33) = 3^{1} \times 7^{1} \times 11^{1} = 3 \times 7 \times 11 = 231$

Answer: 231

d. What is the LCM of 35, 54, and 64?

Prime factors of 35: $5 \times 7$

Prime factors of 54: $2 \times 3^{3} \times (2 \times 3 \times 3 \times 3)$

Prime factors of 64: $2^{6} \times (2 \times 2 \times 2 \times 2 \times 2 \times 2)$

$LCM (35, 54, 64) = 2^{6} \times 3^{3} \times 5^{1} \times 7^{1} = 64 \times 27 \times 5 \times 7 = 60480$

Answer: 60480

Topics related to the Least Common Multiples (LCMs)

Equivalent Ratios

Greatest Common Factors (GCFs)

Operations on Mixed Numbers

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Finding the LCM between numbers provides an opportunity for students to have fun with mathematics, but many learners have a hard time identifying when to use each methodology or try to rush and make small factorization errors. A 1-on-1 tutor can provide extra practice problems and fresh explanations to improve the self-confidence of math students of all ability levels while deepening their understanding of concepts such as LCMs and GCFs. Contact the friendly Educational Directors at Varsity Tutors today to learn more about the potential benefits of private instruction or to sign up.