Math Lesson 1.5.6 - Finding the LCM of Two or More Numbers

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Welcome to our Math lesson on Finding the LCM of Two or More Numbers, this is the sixth lesson of our suite of math lessons covering the topic of Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the LCM of Two or More Numbers

The method used in the last example, that consists of writing the first multiples of two or more numbers, and then identifying the common ones and considering only the least common multiple, is OK but it is a bit long winded and time-consuming. A much easier method is the tabular method discussed in the previous paragraph where we found the prime factors of a number, the difference this time is that it is applied for two or more numbers simultaneously. The division with a certain prime number continues until at least one of the numbers involved is divisible with that prime number. Then, all prime factors written on the right of the vertical line are multiplied with each other to find the least common multiple (LCM) of the given numbers. Let's see an example to clarify this point.

Example 5

Find the least common multiple of 32 and 48.

Solution 5

We use the method described above where the two numbers are written on the left of the vertical line and the prime factors on its right. We have:

Now, we must multiply all factors on the right to find the LCM of 32 and 48. We have

The tabular method can also be applied for 3 or more numbers without any considerable increase in the time required for solving it. Again, we continue the division with a certain prime number until at least one of number is divisible with it. Let's consider another example.

Example 6

Calculate the LCM of 40, 64 and 72.

Solution 6

We have

Thus, we obtain for the least common multiple of the above numbers:

It would have be a very tedious to write all multiples of the three numbers and then look for the least common one as we did earlier for 8 and 12 but the approach is well worth understanding.

The least common multiple of two or more numbers has a wide range of applications in practice. For example, if we want to add or subtract two or more fractions with different denominators, we must find a common denominator first, as a denominator of a fraction represents the way an item is divided. It is obvious that items must be divided in the same way to make them comparable or exchangeable. Hence, we must find a common denominator first, and the best thing to do is to find the least common denominator, as the numbers involved are the smallest possible. For this, we can use the method described above for finding the LCM of two or more numbers.

Another example of LCM used in practice is to find the minimum number of items to use in a situation to reduce costs. Let's see an example.

Example 7

A store sells boxes of pencils, erasers and pens. These boxes contain 20, 24 and 32 items each respectively. What is the minimum number of boxes for each item one teacher must buy if he has to distribute them to his students, where each student must take a set containing a single piece from each item?

Solution 7

First, we must calculate how many items from each category the teacher must buy to complete each student with one of them. Since we are interested about the minimum number possible of items, we have to find the LCM of 20, 24 and 32. We have

Hence, we have for the minimum number of items from each category:

Thus, the minimum number of pencil boxes to purchase is

Likewise, for the minimum number of eraser boxes, we have

Finally, for the minimum number of pen boxes, we have

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7. Continuing learning arithmetic - read our next math tutorial: Divisibility Rules

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