In this video, we will look at some examples of finding the greatest common factor, or GCF. Remember that a number that divides another number evenly is a factor of that number. So, for example, if we wanted to list out the factors of 15, we would write all the numbers that divide 15 evenly. 1 divides 15 evenly, as does 3, 5, and 15. In this video, we're going to be looking for factors that are common to two or more numbers. A number that is a factor of two or more numbers is a common factor of those numbers. So we have the factors of 15 listed. Let's suppose we also were to list the factors of 20. 1 is a factor of 20, as is 2, 4, and 5 are factors of 20. 10 is a factor of 20. And, of course, 20 is a factor of itself. Now we look for the common factors of these two numbers, and we can see that 1 is a common factor of 15 and 20. 5 is also common factor of 15 and 20; and those are the only common factors that we see. Now we're looking for the greatest common factor, so we want the largest common factor of these two numbers. So we can say that 5 is the greatest common factor, or GCF, of 15 and 20. Listing the factors of each number is one way of finding the greatest common factor. But another way to find the greatest common factor is to use the prime factorization of each number. Let's see how that works. We'll start by writing the prime factorization of 15, which is 3 times 5, and then the prime factorization of 20, which is 2 times 2 times 5. So, if we look of the prime factors of 15 and 20, you can see we have a total of three different prime factors: 2, 3, and 5. So let's make a box with three columns: one for the factor 2, one for the factor 3, and one for the factor 5. Now, let's write the prime factorization of each of our numbers in the table, putting the factors in the corresponding column. Starting with 15, we notice that 15 does not have 2 as a factor, so we can leave that column blank in the row for the factorization of 15 or we can put a line there to indicate that 2 is not a factor of 15. 15 has one factor of 3, so we'll put one 3 in that column. And it has one factor of 5. We'll put one 5 there. Now, let's write the prime factorization of 20 in the second row of our table. 20 has two factors of 2, so we will put 2 times 2 in the two column. There are no factors of 3 in 20. So we will leave it blank or put a line in that row in the three column. One factor of 5 in the prime factorization of 20, so we write a 5 in that column. Now we will circle the least product in each column that does not have a blank. Well, there is a blank in the two column. There's a blank in the three column. There is not a blank in the five column. We circled the least factor. There's one factor 5 in each row. It doesn't matter which one we circle. But our greatest common factor is the product of the circled numbers. So that confirms that 5 is the greatest common factor of the numbers 15 and 20. So let's try a couple more examples. Find the greatest common factor of 36, 60, and 72. One way to find the greatest common factor of these three numbers would be to list the factors of each number as we did in the previous example. But another way to find the greatest common factor is to use the prime factorization of each number. So we're going to start by finding the prime factorization of each of our numbers 36, 60, and 72. So we'll start with 36, and 36 is divisible by 2, since it's even. 36 divided by 2 is 18. 18 is even so 2 is a factor of 18. 18 divided by 2 is 9. 2 is not a factor of 9, but 3 is. And 9 divided by 3 is 3. 3 is prime, so we have our prime factorization for 36. It's equal to 2, times 2 times 3 times 3. Now let's look at the prime factorization for 60. 60 is even, so 2 is a factor. 60 divided by 2 is 30. 2 is a factor of 30. 30 divided by 2 is 15. 2 is not a factor of 15, but 3 is. 15 divided by 3 is 5. And 5 is prime. So the prime factorization of 60 is 2 times 2 times 3 times 5. Finally, let's look at the prime factorization for 72. That is an even number, so we will divide it by 2 and get 36. 36 is even, so 2 is a factor of 36. 36 divided by 2 is 18. 18 has 2 as a factor. 18 divided by 2 is 9. 2 is not a factor of 9, but 3 is, and 9 divided by 3 is 3. 3 is prime, so we can stop there. And we see that the prime factorization of 72 is 3 factors of 2 and 2 factors of 3. So that's 2 times 2 times 2 times 3 times 3. Now that we have the prime factorization for each of our numbers, we're going to write the factorization of each number in a table. We have factors of 2, 3, and 5, so we're going to need three columns in our table. So our table will look like this, and we'll start with 36. We're going to write the prime factorization of 36 in this column. 36 has two factors of 2. We'll write those two factors in the two column. Two factors of 3, which we put in the three column. There is no factor of 5 in 36, so we leave that blank, or we can put a dash there. Let's go to our next number, which is 60. 60 has two factors of 2. We'll put those in the two column. One factor of 3 that goes in the three column. And one factor of 5 that goes in the five column. Finally we have the number 72, which has three factors of 2. We put those three factors in the two column, and two factors of 3, which go in the three column. Again, 5 is not a factor of 72, so we'll leave that blank or put a dash there. Now, remember, that to find the greatest common factor, we circled the least product in each column that does not have a blank. Well, the least product in the column two would be 2 times 2. There's two instances of 2 times 2 being the least product. It doesn't matter which one we circle, so we'll circle the first one. In the three column, the least product would be the single factor of 3 in the 60 row. In the five column, we have two blanks: one in the row for the factorization of 36 and one in the row for the factorization of 72, so we don't circle anything there. That tells us 5 is not a common factor to 36 and 72, although it is a factor of 60. Our greatest common factor, then, is going to be the product of those circled numbers, so it's going to be 2 times 2 times 3. That's 4 times 3, which is 12.