# Part 3iv: Least Common Multiple

### Further Discussion:

**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:

**and**

Therefore, we can use to find and vice versa!