1. Factors & Greatest Common Factors
CA 11.0

2. Objective - To write the prime factorization of numbers and find the greatest common factor of monomials.

3. FUNDAMENTAL THEOREM of ARITHMETIC
For every composite number, there is one, and only one, prime factorization.

4. PRIME FACTORIZATION
To find the prime factorization of a composite number, use a
FACTOR TREE
Example:
The prime factorization of 525 is:
3 ● 5 ● 5 ● 7
It can also be written as:
3 ● 52 ● 7
525
5 ● 105
5 ● 5 ● 21
5 ● 5 ● 3 ● 7

5. PRIME FACTORIZATION The prime factorization of 306 is: 306
2 ● 3 ● 3 ● 17
It can also be written as:
2 ● 32 ● 17
306
2 ● 153
2 ● 3 ● 51
2 ● 3 ● 3 ● 17

6. GREATEST COMMON FACTOR (GCF)
The GREATEST COMMON FACTOR (GCF) is the greatest (biggest) number that is a common factor of a group of two or more numbers.

7. GREATEST COMMON FACTOR (GCF)
Example: Find the GCF of 30 and 75.
One way is to list all the factors of 30 and 75.
Now we look for the BIGGEST number in both tables.
The GCF of 30 and 75 is 15.
GCF(30, 75) = 15 30 1 2 15 3 10 5 6 75 1 3 25 5 15

8. GREATEST COMMON FACTOR (GCF)
Example: Find the GCF of 30 and 75.
Another way is to use prime factorization. 30 75
Now we look for all the common factors in both.
The GCF of 30 and 75 is 3 ∙ 5
GCF(30, 75) = 15 3 10 5 15 3 2 5 5 3 5
30 = 2 ∙ 3 ∙ 5
75 = 3 ∙ 5 ∙ 5

9. GCF of a MONOMIAL To find the GCF of a monomial
first find the GCF of the coefficients
then find the GCF for each common variable
The GCF of the common variable will ALWAYS be the variable raised to the SMALLEST EXPONENT

10. The common variable is x and the smallest exponent is 2
GCF of a MONOMIAL
Find the GCF of each pair of monomials:
3x3 and 6x2
The GCF of 3 and 6 is 3
The common variable is x and the smallest exponent is 2
The GCF is:
3x2

11. There are no common variables!
GCF of a MONOMIAL
Find the GCF of each pair of monomials:
4x2 and 5y2
The GCF of 4 and 5 is 1
There are no common variables!
The GCF is: 1

12. GCF of a MONOMIAL Find the GCF of each pair of monomials:
6x3y5 and 15xy9
The GCF of 6 and 15 is 3
The first common variable is x and the smallest exponent is 1
The second common variable is y and the smallest exponent is 5
The GCF is:
3xy5

13. Determine the GCF

14. Factoring
In addition to knowing the GCF, we also want to know what is left over.
Let us look at one of the previous examples:
We already determined that the GCF is
To determine the leftovers DIVIDE each term by the GCF
The leftovers are:

15. Factoring Once we know the GCF and the leftovers, we put them together
GCF is OUTSIDE parenthesis
LEFTOVERS go INSIDE parenthesis
(like you put leftovers inside the fridge)
You CHECK by distributing!
It should match the original!

16. Determine the GCF & the leftovers