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Algebra 2
4-4 Using Prime Factorization
4-4 Using Prime Factorization
 WARMUP
Are the following numbers prime?
37
43
35
42
87
Find a prime number greater than 100.
4-4 Using Prime Factorization
 To factor a number over a set of numbers,
you write it as a product of numbers chosen
from that set – this set will be called the
Factor Set. (We almost always factor to
integers.)
14 could be (1)(14), (-1)(-14), (2)(7) or (-2)(-7)
7, since it is prime, is either (7)(1) or (-7)(-1)
4-4 Using Prime Factorization
 A prime number is an integer greater than 1
whose only positive integral (meaning
integer) factors are itself and 1.
What are the first 10 primes?
2, 3, 5, 7, 11, 13, 17, 19, 23, and 29
4-4 Using Prime Factorization
 Prime Factorization:
Systematically, try the primes, in order, as
factors. Use each repeatedly until it is no
longer a factor.
4-4 Using Prime Factorization
 Example:
Find the Prime Factorization of 600
600 = 2 · 300
= 2 · 2 · 150
= 2 · 2 · 2 · 75
= 2 · 2 · 2 · 3 · 25
=2·2·2·3·5·5
= 2 3 · 3 · 52
4-4 Using Prime Factorization
 Greatest Common Factor (GCF) of two or
more integers is the greatest integer that is a
factor of both.
 Least Common Multiple (LCM) of two or more
integers is the least positive integer having
each as a factor.
When given two or more integers, you can use
their prime factorization to find their GCF and
LCM.
4-4 Using Prime Factorization
 Find the GCF of:
72, 108 and 126
First find the prime factorizations of the 3
numbers…
4-4 Using Prime Factorization
 72 = 23  32
 108 = 22  33
 126 = 2  32  7
To find the GCF, take the least power of each
common prime factor.
 GCF = 2  32 = 18
4-4 Using Prime Factorization
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Now find the LCM.
72 = 23  32
108 = 22  33
126 = 2  32  7
To find the LCM, take the greatest power of
each prime factor.
 LCM = 23  33  7 = 1512
4-4 Using Prime Factorization
 Let’s do a simpler one:
Find the GCF and the LCM of 24 and 16:
24 = 23 · 3
16 = 24
To find the GCF, take the least power of each
COMMON prime factor. In this case, just 23.
GCF = 8
To find the LCM, take the greatest power of
EACH prime factor. In this case, 24 and 3.
LCM = 24 · 3 = 48
4-4 Using Prime Factorization
 GCF and LCM can apply to polynomials as
well.
Look at example at bottom of page 180
4-4 Using Prime Factorization
 What is the GCF and the LCM of:
18 and 20
4-4 Using Prime Factorization
4-4 Using Prime Factorization
 Do more examples.
4-4 Using Prime Factorization
 HOMEWORK
p. 181 # 9-21 ALL
TEST on Thursday! Ch. 3