Least Common Multiples
Another common task that you'll come across, especially when working with fractions, is finding the least common multiple of two numbers. There are several very close parallels that we'll see between least common mutiples and the greatest common factors that we discussed in the previous lesson.
a is a common multiple of b and c if a is a multiple of both b and c. a is the least common multiple of the two numbers if there are no common multiples smaller than a.
We'll write "the least common multiple of a and b" as lcm(a, b).
Both phrases mean exactly what they seem to say: A number is a common multiple of two other numbers if it's a multiple of both of them. So, for example, 36 is a common multiple of 3 and 4 but 8 isn't a common multiple since it's a multiple of 4 only. A little experimentation will show you that 12 and 24 are also multiples that 12 and 18 have in common. Because 12 is the smallest of the common multiples, it would be the least common multiple of 3 and 4.
The procedure for finding the least common multiple is very similar to what we used to find the greatest common factor.
- Write out the prime factorizations of both numbers.
- List all of the factors even if they only appear in one of the numbers.
- Give each factor the greatest exponent from the two factorizations.
- Multiply the resulting numbers together.
I put the differences between this method and the one for finding the greatest common factor in italics. With the least common multiple, you need all of the factors, not just the ones that they have in common, and you take the largest exponent from the two prime factorizations rather than the smallest.
Example 1
Find the least common multiple of 100 and 80.
| 1. Write the prime factorizations of each number. | 100 = 22 · 52 80 = 24 · 5 |
| 2. List all the factors from both numbers. | 2, 5 |
| 3. Take the greatest exponent from each number. | 24, 52 |
| If we multiply those numbers together, we'll get our least common multiple. | gcf(100, 80) = 24 · 52 = 400 |
Example 3
Find lcm(495, 945)
We can also use a table-type method to find least common multiples that's similar to the one we used for greatest common factors. I'm going to start by writing out the prime factorizations with the common factors lined up.
| 495 | = | 3 | · | 3 | · | 5 | · | 11 | ||||
| 945 | = | 3 | · | 3 | · | 3 | · | 5 | · | 7 | ||
| gcf(495, 945) | = | 3 | · | 3 | · | 3 | · | 5 | · | 7 | · | 11 |
| gcf(495, 945) | = | 10395 |
To find the least common multiple, you use the numbers in every column even if the number appears in only one row.
Example 2
Find the least common multiple of 81 and 54.
| 1. Write the prime factorizations of each number. | 81 = 34 27 = 32 · 2 |
| 2. List all the factors from both numbers. | 3, 2 |
| 3. Take the greatest exponent from each number. | 34, 21 |
| If we multiply those numbers together, we'll get our least common multiple. | gcf(81, 54) = 34 · 21 = 162 |
Example 4
Find the least common multiple of 1080 and 108
We'll do this one using the table method.
| 1080 | = | 2 | · | 2 | · | 2 | · | 3 | · | 3 | · | 3 | · | 5 |
| 108 | = | 2 | · | 2 | · | 3 | · | 3 | · | 3 | ||||
| gcf(1080, 108) | = | 2 | · | 2 | · | 2 | · | 3 | · | 3 | · | 3 | · | 5 |
| gcf(1080, 108) | = | 1080 |
Videos
Dynamic Tutorial - Finding the Least Common Multiple
Directions: This solution has 4 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.
