Greatest Common Factor (GCF)

In today's lesson we learned three things: GCF of numbers, relatively prime, and the GCF of monomials. As long as we have good notes and can follow the steps we should be able to find the GCF of both numbers and monomials.

Let's start with the GCF of numbers. We talked in class about 2 different ways to find the GCF of numbers. The first way is to list the factors of each number. This works really well if we have small numbers. Here's an example:

16: 1, 2, 4, 8, 16

32: 1, 2, 4, 8, 18, 32

40: 1, 2, 4, 5, 8, 10, 20, 40

Now that the factors are listed we can find the largest one that they have in common and that is our answer. So the GCF is 8.

The second way, which works well if we have large numbers, is to find the prime factorization of each number and then find the common prime numbers and multiply them. Here's what I mean using the same example from above (now I can't make a factor tree on this website, so I have skipped the step and already listed the prime numbers).

16: 2 * 2 * 2 * 2

32: 2 * 2 * 2 * 2 * 2

40: 2 * 2 * 2 * 5

We can see that they each have three 2's in common (2 * 2 * 2). So if I multiply them together I get 8.

Next we discussed the term relatively prime. Two or more numbers are relatively prime if their greatest common factor is one. If the only factor they have in common is one they are relatively prime.

Finally we discussed finding the GCF of monomials. Here are the steps that we wrote down in class:

1. Factor the monomials (see last weeks notes if you need help)
2. Find all the common factors (circling them helps)
3. Your answer is all common factors multiplied together (everything you circled)

Here is an example (Remember we discussed the look of exponents on here during class):

12m2n3: 2 * 2 * 3 * m * m * n * n * n

70m3n: 2 * 5 * 7 * m * m * m * n

We notice that these monomials have a 2, m, m, and n in common. If I multiply those together I get the answer of 2m2n