Part 3iv: Least Common Multiple
GCF and LCM, A More Indepth Look
If you would like to take the discussion of GCF and LCM one step further, there is a nice relationship between whole numbers and , , and .
Let’s use the same numbers we used in the previous clip, and . If we use exponential form, we can express the prime factorizations of these numbers as follows:
Now the procedure for computing is to take each prime the smaller number of times it appears in any of the prime factorizations, as these will be common factors. To obtain the greatest common factor, we multiply all of these powers of primes together:
Since the prime factor occurs once in the prime factorization of and twice in the prime factorization of , we take one factor of when computing the GCF (as only one factor is common). Similarly, we take no factors of , one factor of and no factors of .
But when we compute , we take each prime the greater number of times that it appears in any of the prime factorizations. When we multiply all of these powers of primes together, we will obtain the least common multiple:
Since the prime factor occurs once in the prime factorization of and twice in the prime factorization of , when building the LCM we need to take two factors of (if we take fewer than two factors of the number we eventually build will not be a multiple of both and , and if we take more than two factors of the number we eventually build will not be the least common multiple). Similarly, we need one factor of , two factors of , and one factor of .
Now comes the interesting part, and we will see why some of the above numbers are green while some others are red. Multiply and , but do so with the prime factorizations, and notice that we can use the Any-Order property of multiplication to multiply numbers in any order we like:
There is nothing special about the numbers and ; for any two whole numbers and , the following relationship holds (the general proof is based on the above explanation):
So if we manipulate this equation ever so slightly, we obtain the following equations:
Therefore, we can use to find and vice versa!