In the first reading, we talked about factor trees to find the prime factorization of a number. Here's an example of a factor tree:
 

In this reading, we'll talk about how to use prime factorizations to find the Greatest Common Factor, or GCF of two numbers. 

Do you know what a GCF is? The best way to understand it is by example. The GCF of 12 and 16 is 4. Why? Because 4 is the biggest (greatest) number which is a factor of both 12 and 16. Both 12 and 16 are divisible by 4, and there is no number bigger than 4 which both 12 and 16 can be divided by.

So how do we use prime factorizations to find the GCF of two numbers? I'll show you.

Example 1: Find the GCF of 24 and 30.

Solution: If you did a factor tree on the two numbers you would find that:

24 = 2³ x 3
30 = 2 x 3 x 5

Just a reminder, if there is no exponent with a prime factor, it's understood to be 1, so we could also write the factorizations like this:

24 = 2³ x 3¹
30 = 2¹ x 3¹ x 5¹

Here's what we do next: We ask ourselves, "What factors appear in both factorizations?"

The answer is: 2 and 3. 5 appears in one of them, but not both.

Now the next question we ask is: "What is the smallest exponent for each prime factor?"

For 2, the smallest exponent is a 1, and for 3 it is also 1.

Thus, the GCF is 2¹ x 3¹ = 6.

Example 2: What is the GCF of 16 and 96?

Solution: The prime factorizations of 16 and 96 are:

16 = 2⁴
96 = 2⁵ x 3

Only 2 appears in both factorizations, and the smallest exponent for 2 is four, so the GCF = 2⁴ = 16.

Example 3: What is the GCF of 24 and 72?

Solution: The prime factorizations of 24 and 72 are:

24 = 2³ x 3
96 = 2³ x 3²

Both 2 and 3 appear in both factorizations. The smallest exponent for 2 is three, and the smallest exponent for 3 is 1. So the GCF = 2³ x 3 = 24.

Example 4: What is the GCF of 8x³y⁴z and 12x²y⁷?

Solution: The Prime factorizations are:

2³x³y⁴z 
2²3x²y⁷

The result has to contain 2, x and y, but not z(because only one of the factorizations has a z in it.)

The result is 2²x²y⁴, or 4x²y⁴