Greatest Common Factor!!

Okay, so first let's define what a greatest common factor (GCF) is: the GCF of two or more numbers is the largest number that divides into both numbers. We are going to look at three simple ways to find the GCF: Venn diagrams, prime factorization, and the Euclidean Algorithm.

So we all remember Venn diagrams right? A Venn Diagram is a tool used to sort and compare information. When it comes to GCF, we are simply sorting the factors for two or more numbers.  For this first example, we are going to factor 36 and 42.  In the left circle we will list the factors of 36, and in the right circle we will list the factors of 42.  In the intersection, we will list the factors that are Common to both 36 and 42.

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.  The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.  The factors Common to both 36 and 42 are: 1, 2, 3, 6.  In this visual representation, it is easy to find the greatest, or largest, common factor.  We can see that the GCF is 6.





A second method to find the GCF is prime factorization.  With prime factorization, we use 'factor trees' to find which prime numbers we need to multiply to get a number.  For example, the prime factorization of 36 is: 2x2x3x3.  Let's use the numbers 24 and 72 for this example.

With these factor trees, we want to find two factors that multiply together to get the original number.  2x12=24.  However, we want to keep doing this until all factors are primes, so we have to factor 12, and then 4 and so on. In the end we find that: 24=2x2x2x3 and 72=2x2x2x3x3.  Now that we have the prime factorization, we want to find the "common multiples".  In this case, the common multiples are 2x2x2x3; after we multiply these, we find that the GCF is 24!  

A third method to finding the GCF is by using the Euclidean Algorithm.  For this algorithm, I found a great video that goes through each step in detail! Let's watch!

Well that is it for today.  For more examples and more practice on finding the GCF, click here!