Factors & Greatest Common
Factors
CA 11.0
Objective - To write the prime
factorization of numbers and
find the greatest common factor
of monomials.
FUNDAMENTAL THEOREM
of ARITHMETIC
For every composite number, there is
one, and only one, prime factorization.
PRIME FACTORIZATION
To find the prime factorization of a composite
number, use a
FACTOR TREE
Example:
525
5 ● 105
The prime factorization
of 525 is:
3●5●5●7
5 ● 5 ● 21
It can also be written as:
5 ● 5 ● 3●7
3 ● 52 ● 7
PRIME FACTORIZATION
306
2 ● 153
The prime factorization
of 306 is:
2 ● 3 ● 3 ● 17
2 ● 3 ● 51
It can also be written as:
2 ● 3 ● 3 ● 17
2 ● 32 ● 17
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.
GREATEST COMMON FACTOR
(GCF)
Example: Find the GCF of 30 and 75.
One way is to list all the factors of 30 and 75.
30
1 30
2 15
3 10
5 6
75
1 75
3 25
5 15
Now we look for the
BIGGEST number in
both tables.
The GCF of 30 and
75 is 15.
GCF(30, 75) = 15
GREATEST COMMON FACTOR
(GCF)
Example: Find the GCF of 30 and 75.
Another way is to use prime factorization.
30
3
3
75
10
2
5
5
30 = 2 ∙ 3 ∙ 5
75 = 3 ∙ 5 ∙ 5
5
Now we look for all
the common factors
in both.
15
3
5
The GCF of 30 and
75 is 3 ∙ 5
GCF(30, 75) = 15
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
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
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
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
Determine the GCF
6 x  3x
5
3
25 x yz  70 x z
2
4
4
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:
6 x  3x
5
3
We already determined that the GCF is
3x
3
To determine the leftovers DIVIDE each
term by the GCF
5
3
6 x 3x
The leftovers are: 2 x 2  1
3x
3

3x
3
Factoring
Once we know the GCF and the leftovers,
we put them together


3x 2 x  1
3
GCF is OUTSIDE
parenthesis
2
LEFTOVERS go INSIDE
parenthesis
(like you put leftovers inside
the fridge)
You CHECK by distributing!
It should match the original!
6 x  3x
5
3
Determine the GCF & the
leftovers
12 x y  8x y
4
3
7
25 x yz  70 x z
2
4
4