We have been using techniques to find factors of numbers including the greatest common factor. Previously we have used a Venn Diagram. Below is an example.
Yesterday we used the prime factorization technique. An example can be found below.
Basically, you find the prime factorization of both numbers. We used the factor tree method. The prime factorization of 24 is 2 x 3 x 2 x 2. The prime factorization of 60 is 5 x 2 x 3 x 2. The factor string in bold is what they have in common. By multiplying the common factor string of 2 x 3 x 2, we can find the greatest common factor of 24 and 60, which is 12.
We also can find the least common multiple of a number, or LCM by doing the same thing. First lets look at the previous method. This method is the listing method, which we just list the multiples by multiplying the original number by 1, 2, 3, etc. until we find common multiples. In this case the LCM of 24 and 60 is 120.
By using the factor tree method and finding the prime factorization of both 24 and 60, we can see that 3 x 2 x 2 x 2 is the prime factorization of 24 and 5 x 2 x 3 x 2 is the prime factorization of 60.
When finding the LCM, we still use the factor string that they have in common, 2 x 3 x 2, but we add to it the factors they don't have in common. If we take the new factor string that combines what they have in common and what they don't have in common ,2 x 2 x 2 x 3 x 5, we get 120.
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