BMCC: Math Tutorials: Numbers:
Least Common Multiple
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Multiples
Common Multiples
Least Common Multiple
Before defining the least common multiple, let's review how we define multiples. For any number n, a multiple of n is a product obtained when n is multiplied by any whole number. For instance, if we start with the number 7, then 28 is a multiple of 7 because 7 x 4 = 28. Another multiple of 7 is 42, because 7 x 6 = 42. The number 39 would not be a multiple of 7, as there is no whole number by which 7 can be multiplied to yield the product 39. However, 39 is a multiple of 3, since 3 x 13 = 39.
When we speak of the multiples of a number, we mean the entire list of multiples that exist for that number. There is always an infinite number of multiples for every number, so we usually just list the first few. For a number n, then, a complete list of multiples would start out as the products of n x 1, n x 2, n x 3, n x 4, n x 5, . . . and continue forever. Remember that every number is a multiple of itself, because of n x 1.
As an example, consider the number 6. The multiples of 6 are:
Products that are multiples of more than one number are common multiples for all of their factors. If you think about it, any number that is not a prime number has to be a common multiple for how ever many factors it has. Consider the number 30. We can show that the factors for 30 are 1, 2, 3, 5, 6, 10, 15, and 30. 30 is a multiple of each of these factors (we usually don't bother including 1 in the list of multiples), so we say that 30 is a common multiple for the numbers 2, 3, 5, 6, 10, 15, and 30.
Suppose we want to determine the common multiples for two numbers. We first make out a list of multiples for each number, listing a few in each case. Then we compare the lists and see if any numbers appear on both lists. Any number that does appear on both lists is a common multiple for those two numbers. For example, determine some common multiples for the numbers 4 and 6. The beginnings of the lists of multiples for each number look like this:
| 4 | 6 | |
| 4 | 12 | |
| 8 | 18 | |
| 12 | 24 | |
| 16 | 30 | |
| 20 | 36 | |
| 24 | 42 | |
| 28 | 48 | |
| 32 | 54 | |
| 36 | 60 |
To determine some common multiples, simply look at which multiples appear in both lists. In this case, we can see that 12, 24, and 36 are common multiples of the numbers 4 and 6. You can never list all of the common multiples for any two or more numbers, as there are an infinite number of common multiples, just like each number has an infinite number of multiples.
If you think about it, any two numbers (this works for three or more numbers too) have to have common multiples. At the very least, you can find a common multiple by multiplying the two numbers together. Consider the numbers 3 and 7. A list of multiples for 3 starts out as 3, 6, 9, 12, 15, 18, 21, 24, . . . and a list of multiples for 7 starts out as 7, 14, 21, 28, . . . The first common multiple that we see is 21, which is the product of 3 x 7. Thus, you can find a common multiple for any two numbers simply by multiplying them together. The resulting product has to be a common multiple, although it will not always be the first one you would find by comparing complete lists of multiples for the two numbers.
Now we finally come to the title topic of this section. The least common multiple for two or more numbers is the smallest number in the list of common multiples for the numbers. Once you already have a list of common multiples generated like we did above, the least common multiple can be determined by just looking at the list. In the example of the numbers 4 and 6 that we did earlier, the least common multiple of those two numbers is 12. You can determine the least common multiple for any two or more numbers by making a list of multiples for each, and seeing what the smallest number is that shows up on all of the lists-- that number will be the least common multiple.
There's another way to determine the least common multiple for numbers: using prime factorization. Making lists of multiples and doing a visual inspection is a handy way to determine the least common multiple for two numbers, but it starts to get cumbersome if you are trying to determine the least common multiple for three or more numbers. Sometimes it's quicker to find the least common multiple by first figuring the prime factorization for each number.
Here's how it works. Suppose we want to find the least common multiple for the numbers 6 and 20. First, we find the prime factorization for each number as described in the previous section. The prime factorization for 6 is 2 x 3, and the prime factorization for 20 is 2 x 2 x 5. Next we want to make a table, with one column for each different prime factor for the two numbers, and one row for each of the numbers. By different prime factors, I mean each prime factor that is found for either of the two numbers. In this case, the different prime factors are 2, 3, and 5 (2 is a prime factor for both 6 and 20, 3 is a prime factor for only 6, and 5 is a prime factor for only 20). For each number on the left-hand side of the table, the number of times that the prime factor goes into the number is entered.
The table looks like this for 6 and 20:
| 2 | 3 | 5 | |
| 6 = | 2 | 3 | 0 |
| 20 = | 2 x 2 | 0 | 5 |
To determine the least common multiple, multiply together the largest values that are entered within each prime factor column. The largest value in the column for prime factor 2 is 2 x 2, the largest value in the middle column is 3, and the largest value in the right-hand column is 5. Here's the same table again, with the entries of interest highlighted with green background:
| 2 | 3 | 5 | |
| 6 = | 2 | 3 | 0 |
| 20 = | 2 x 2 | 0 | 5 |
Multiplying all of these together gives
Therefore the least common multiple for the numbers 6 and 20 is 60.
That was a fairly simple example, and you probably could have figured out pretty quickly in your head that the least common multiple was going to be 60. The advantage of the prime factorization method, though, is that it works for more complicated problems too. Suppose I want to know the least common multiple for the three numbers 15, 28, and 126. This one might be a little tough to do in your head! But the approach is the same as for the problem we just did. First we figure out the prime factorization for each number. You can verify each of these results as a review of the last section.
The different prime factors for the three numbers are 2, 3, 5, and 7. The table looks like this:
| 2 | 3 | 5 | 7 | |
| 15 = | 0 | 3 | 5 | 0 |
| 28 = | 2 x 2 | 0 | 0 | 7 |
| 126 = | 2 | 3 x 3 | 0 | 7 |
The largest values in each of the columns are (from left to right) 2 x 2, 3 x 3, 5, and 7. If one column has more than one cell with the greatest value, like the right-hand column having two cells with the value 7, you should still only use the value from one of the cells, not both. Here's the same table again, with the cells of interest highlighted with green background:
| 2 | 3 | 5 | 7 | |
| 15 = | 0 | 3 | 5 | 0 |
| 28 = | 2 x 2 | 0 | 0 | 7 |
| 126 = | 2 | 3 x 3 | 0 | 7 |
Multiplying all of these together gives:
Thus, the least common multiple for the numbers 15, 28, and 126 is 1260!
Next: Greatest Common Factor
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