Transcript Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Greatest Common Factor
(GCF) and Least Common
Multiple (LCM)
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
2 is prime.
Because
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
Factoring Monomials
You can use prime factorization to factor 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
first.
100
10
5
Now expand
the variable
mn = mnnn
expression.
10
2 5
3
2
2 255
Combine the results.
2  2  5  5mnnn
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)
The Greatest Common Factor (GCF) 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.
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 (LCM)
The Least Common Multiple (LCM) is the
least number that is a common multiple of
two or more 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 GCF 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).