GCF and LCM... and fractions!!! Mwahahaha!

Hello all. It's been a while.

Let's review how we can find the GCF (greatest common factor) and the LCM (least common multiple) of each set of numbers by using the prime factorization method.

1)  16, 60

First, find the prime factorization of each:

Next, identify the matches on each side. A "match" is when there's a number on one side (at the bottom of the tree) that has an EXACT twin on the other side. The two numbers team up to create a "match."

We have underlined our matches. There's a "2" on one side, and another "2" on the other side. This gives us our first match: 2. Our second match is also 2, since there is a second two on both sides. As shown in the picture below, the remaining numbers have no matches:

Now, list our matches. We have one match of "2" and another match of "2". We multiply our matches together:

The GCF = 4.

***Please note - if you are finding the GCF of more than two numbers, a number must be present for ALL the prime factorizations in order to be considered a match.***


As for the LCM...

First, find the prime factorization of the two numbers:

               8                   18
           2 x 4               2 x 9
        2  x 2 x 2         2 x 3 x 3

Our prime factorizations are at the bottom. Now, we need to multiply all of the bottom "tree" numbers together, using the matches ONLY ONCE. Our only match between the two tree bottoms is a pair of twos (in red), so we use it only once, and then bring all the other numbers down to multiply:

2 x 2 x 2 x 3 x 3 = 72

The LCM of 8 and 18 is 72.

Let's try it with three numbers. Find the LCM of 6, 14, and 28 using prime factorization:

First, find the prime factorization of all three numbers (I'm going to skip the whole tree and go straight to the answers we'd have at the bottom):

      6                       14                      28
   2 x 3                   2 x 7               2 x 2 x 7    

We have a matching 2 for all three trees, so we use it once. We also have a match of 7 between 14 and 28, so we use it only once. Then, we multiply those with all of the leftover, non-matching numbers:

2 x 7 x 3 x 2 = 84. The LCM of 6, 14, and 28 is 84.

***Please note - if you are finding the LCM of more than two numbers, a number need only be present in TWO of the prime factorizations in order to be considered a match.***

Now. What on earth does all this LCM stuff have to do with fractions? Remember, in ancient times (a couple of days ago) when we had to compare fractions, that we used the LCM of the denominators (the "LCD" - least common denominator) to turn fractions with different denominators into fractions with the same denominators. (You can't compare two fractions that have different denominators.) Kinda like giving them the same last name, or putting them into the same "family."

In order to compare fractions with unlike denominators, you must:

1) Find the LCD.
2) Convert the original fractions into equivalent fractions using the LCD.
3) Compare the numerators, and voila!

Let's walk through an example using these steps:

Which is greater, 2/3 or 7/8? (Sorry about the sideways fractions!)

1) Find the LCM of the denominators:
          3                   8
          3               2 x 2 x 2 
We have no matches, so we multiply everything together: 3 x 2 x 2 x 2 = 24. This is now the Least Common Denominator we're looking for. 

2) Convert the original fractions into equivalent fractions with the Least Common Denominator.
That means we need to change 2/3 and 7/8 into fractions that have 24 at the bottom.

2/3 = ?/24 --- First, figure out how we can get from 3 to 24. Once we realize we multiply by 8, we must do the same thing to the numerator. Our new fraction is 16/24.

7/8 = ?/24 --- How do we get from 8 to 24? We multiply by 3. Do the same thing to the top for our new fraction: 21/24.

3) Now we can finally compare the two fractions by simply looking at the numerator. Which is bigger, 16/24 or 21/24?
We know that 21/24 is bigger. Therefore, 7/8 is greater than 2/3. 


And finally... some food for thought from my favorite internet friend, Philosoraptor: