How to Find the Least Common Multiple

The least common multiple (LCM) of a set of numbers is the smallest number that’s a multiple of every number in that set. For small numbers, you can simply list the first several multiples of each number until you get a match.

When you’re finding the LCM of two numbers, you may want to list the multiples of the larger number first, stopping when the number of multiples you’ve written down equals the smaller number. Then list the multiples of the smaller number and look for a match.

However, you may have to write down a lot of multiples with this method, and the disadvantage becomes even greater when you’re trying to find the LCM of large numbers. Try a method that uses prime factors when you’re facing big numbers or more than two numbers. Here’s how:

1. Write down the prime decompositions of all the numbers.

2. For each prime number you find, underline the most repeated occurrences of each.

   In other words, compare the decompositions. If one breakdown contains two 2s and another contains three 2s, you’d underline the three 2s. If one decomposition contains one 7 and the rest don’t have any, you’d underline the 7.

3. Multiply the underlined numbers to get the LCM.

Sample questions

1. Find the LCM of 6 and 8.

   24. Because 8 is the larger number, write down six multiples of 8:

   Multiples of 8: 8, 16, 24, 32, 40, 48

   Now, write down multiples of 6 until you find a matching number:

   Multiples of 6: 6, 12, 18, 24

2. Find the LCM of 12, 15, and 18.

   180. Begin by writing the prime decompositions of all three numbers. Then, for each prime number you find, underline the most repeated occurrences of each:

   12 = 2 x 2 x 3

   15 = 3 x 5

   18 = 2 x 3 x 3

   Notice that 2 appears in the decomposition of 12 most often (twice), so underline both of those 2s. Similarly, 3 appears in the decomposition of 18 most often (twice), and 5 appears in the decomposition of 15 most often (once). Now, multiply all the underlined numbers:

   2 x 2 x 3 x 3 x 5 = 180

Practice questions

1. Find the LCM of 4 and 10.

2. Find the LCM of 7 and 11.

3. Find the LCM of 9 and 12.

4. Find the LCM of 18 and 22.

Following are the answers to the practice questions:

1. The LCM of 4 and 10 is 20.

   Write down four multiples of 10:

   Multiples of 10: 10, 20, 30, 40

   Next, generate multiples of 4 until you find a matching number:

   Multiples of 4: 4, 8, 12, 16, 20

2. The LCM of 7 and 11 is 77.

   Write down seven multiples of 11:

   Multiples of 11: 11, 22, 33, 44, 55, 66, 77

   Next, generate multiples of 7 until you find a matching number:

   Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77

3. The LCM of 9 and 12 is 36.

   Write down nine multiples of 12:

   Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108

   Next, generate multiples of 9 until you find a matching number:

   Multiples of 9: 9, 18, 27, 36

4. The LCM of 18 and 22 is 198.

   First, decompose both numbers into their prime factors. Then underline the most frequent occurrences of each prime number:

   18 = 2 x 3 x 3

   22 = 2 x 11

   The factor 2 appears only once in any decomposition, so I underline a 2. The number 3 appears twice in the decomposition of 18, so underline both of these. The number 11 appears only once, in the decomposition of 22, so underline it. Now, multiply all the underlined numbers:

   2 x 3 x 3 x 11 = 198