Prime Factorization, Greatest
Common Factor (GCF), and
Least Common Multiple (LCM)
Definition of a Prime Number
A prime number is a whole number greater
than 1 AND can only be divided evenly by 1
and itself. Examples are 2, 3, 5, 7, and 11.
Example: Are the following numbers prime?
Why or why not?
a) 31
b) 1 Yes, 31 is greater than 1 AND is only divisible by 1 and 31.
c) 51 No, 1 is not greater than 1.
No, 51 is divisible by 1, 3, 17 and 51. Not just 1 and 51.
Prime Factorization
Prime Factorization is when a number is expressed
as a product of prime factors.
You can use a factor tree to find the prime
factorization of a number.
Example: Find the prime factorization of 12.
12
Because
2 is prime.
2  6  12 2
Bring it down
on the same level.
Notice this equals 12.
2
6
2
2 23
Because
23  6
3
Only Primes
numbers are left
Prime Factorization of Algebraic Monomials
You can use prime factorization to factor algebraic
monomials. A monomial is in factored form when it is
expressed as the product of prime numbers and variables
and no variable has an exponent greater than 1.
Example: Factor 100mn3.
Factor 100
Now expand
100
the variable
first.
10
5
mn = mnnn
expression.
10
2 5
3
2
2 255
Combine the results.
2  2  5  5mnnn
Definition of Factors of a Number
The factors of a number are all the numbers
that can be divided evenly into the number
with no remainder. For instance 5 is a factor of
30 because 30 divided by 5 is 6 (a whole
number). When finding factors, we typically
focus on whole numbers.
Example: Find a factor of 48.
All the factors of 48 are: 1, 2, 3, 4, 6, 8, 12,
16, 24, 48.
Definitions for Common Factors of Numbers
Factors that two or more numbers have in
common are called common factors of those
numbers.
Example: Find a common factor of 24 and 30.
All the common factors of 24 and 30 are 1, 2, 3,
and 6.
The GCF of 24 and
30 is 6.
The largest common factor of two or more
numbers is called their Greatest Common Factor
(GCF).
Finding a GCF through Reasoning
Find the greatest common factor of:
8, 28, 12
Factors of 8:
1
2
4
8
Factors of 28:
1
2
4
7
14
28
Factors of 12:
1
2
3
4
6
12
Find the largest number that divides all three numbers.
The GCF is 4.
This process could be long and tedious if the numbers are
large. A more efficient method is desirable.
Greatest Common Factor (GCF)
for Algebraic Expressions
The Greatest Common Factor (GCF) for
algebraic expressions is the greatest
expression that is a factor of the original
expressions.
Procedure to find the GCF of two or more
terms:
1. Factor each monomial.
2. The GCF is the product of the common
factors.
Examples
Find the GCF of each set of monomials:
a) 24, 60, and 72
24  2  2  2  3
60  2  2  3  5
72  2  2  2  3  3
b) 15 and 8
15  3  5
8  222
2  2  3  12
There is nothing
in common, so
the GCF is…
c) 15a2b, 9ab2, and 18ab
15a b  3  5 a a b
2
9ab  3  3 a b b
18ab  3  2  3 a b
2
3ab
1
Step One: Factor
each Monomial.
Step Two: Find
the common
factors.
Step Three:
Multiply the
factors in
common.
Definitions for Multiples of Numbers
A multiple is the result of multiplying a number by an
integer.
A common multiple is a number that is a multiple of
two or more numbers.
Example: Find common multiples of 3 and 4.
The common multiples of 3 and 4 are 12, 24, 36, 48, etc.
The LCM of 3 and 4 is 12.
The smallest common multiple of two or more numbers
is called their least common multiple (GCF).
Finding a LCM through Reasoning
Find the least common multiple of:
8, 28, 12
Multiples of 8:
Multiples of 28:
Multiples of 12:
8
16
24
32
40
48
56
64
72
80
88
96
104
112
120
128
136
144
152
160
168
176
184
192
28
56
84
112
140
168
196
224
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
Find the smallest number that is a multiple of all three.
The LCM is 168.
This process could be long and tedious if the numbers are
large. A more efficient method is desirable.
Least Common Multiple for Algebraic
Expressions (LCM)
The Least Common Multiple (LCM) for algebraic
expressions is the least expression that is a
common multiple of the original
expressions.
Procedure to find the LCM of two or more terms:
1. Factor each monomial.
2. Find the greatest number of times each
factor appears in each factorization.
3. The LCM is the product of (2).
Examples
Find the LCM of each set of monomials:
a) 18, 30, and 105
18  2  3  3
30  2  3  5
105  3  5  7
Step One: Factor
each Monomial.
2  3  3  5  7  630
b) 15a2b and 27b3
15a 2b  3  5 a a b
27b3  3  3  3 b b b
5  3  3  3aabbb
2 3
 135a b
Step Two: Find
the greatest
number of times
each factor
appears in each
factorization.
Step Three:
Multiply the
result of (2).