Transcript 7.1 Factors and Greatest Common Factors (GCF) CORD Math Mrs. Spitz Fall 2006 Objectives • Find the prime factorization of an integer, and • Find the.

7.1 Factors and Greatest
Common Factors (GCF)
CORD Math
Mrs. Spitz
Fall 2006
Objectives
• Find the prime factorization of an integer, and
• Find the greatest common factor (GCF) for a
set of monomials.
Assignment
• pp. 258-259 #17-22, 29-32, 37-63 (every 3rd
problem) and 71.
Connection
• In mathematics, there are
many situations where there
is more than one correct
answer. Suppose three
students are asked to draw a
rectangle that has an area of
18 square inches. As shown,
each student can draw a
different rectangle, and each
rectangle is correct.
2 in.
9 in.
6 in.
18 in.
1 in.
3 in.
Factors/prime numbers and composite numbers
• Since 2 x 9, 3 x 6, and 1 x 18
all equal 18, each rectangle
has an area of 18 square
inches. When two or more
numbers are multiplied, each
number is a factor of the
product. In the example
given, 18 is expressed as the
product of different pairs of
whole numbers.
18 = 2 · 9
18 = 3 · 6
18 = 1 · 18
• The whole numbers 1,
18, 2, 9, 3, and 6 are
factors of 18. Some
whole numbers have
exactly two factors, the
number itself and1.
These numbers are
called prime numbers.
Whole numbers that
have more than two
factors are called
composite numbers.
Definition of Prime and Composite
Numbers
• A prime number is a
whole number, greater
than 1, whose only
factors are 1 and itself.
• A composite number is
a whole number, greater
than 1, that is not
prime.
0 and 1 are neither prime nor
composite
• The number 9 is a factor of 18, but not a
prime factor of 18, since 9 is not a prime
number. When a whole number is expressed
as a product of factors that are all prime, the
expression is called prime factorization of
the number. Thus the prime factorization of
18 is 2 · 3 · 3 or 2 · 32 .
What else?
• The prime factorization of every number is
unique except for the order in which the
factors are written. For example, 3 · 2 · 3 is
also a prime factorization of 18, but it is the
same as 2 · 3 · 3. This property of numbers
is called the Unique Factorization Theorem.
Ex. 1: Find the prime factorization of 84.
• You can begin by dividing 84 by its least prime factor.
Continue dividing by least prime factors until all the
factors are prime.
84  2  42 The least prime factor of 84 is 2.
 2  2  21 The least prime factor of 42 is 2.
 2  2  3  7 The least prime factor of 21 is 3.
All of the factors in the last row are prime. Thus, the prime
factorization of 84 is 2 · 2 · 3 · 7 or 22 · 3 · 7
Ex. 2: Factor -525.
• To factor a negative integer, first express it as the
product of a whole number and -1. Then find the
prime factorization.
 525  1 525
 1 3 175
 1 3  5  35
 1 3  5  5  7
Take the integer and take out the -1.
The least prime factor of 525 is 3.
The least prime factor of 175 is 5.
The least prime factor of 35 is 5.
All of the factors in the last row are prime. Thus, the prime
factorization of -525 is -1 · 3 · 5 · 5 · 7 or -1 · 3 · 52 · 7
Ex. 3: Factor 20a2b.
• A monomial is written in factored form when it is expressed as
the product of prime numbers and variables where no variable
has an exponent greater than 1.
20a 2b  2 10  a  a  b The least prime factor of 20 is 2.
 2  2  5  a  a  b The least prime factor of 10 is 2.
All of the factors in the last row are prime. Thus, the prime
factorization of 20a2b is 2 · 2 · 5 · a · a · b or 22 · 5 · a · a · b
Two or more numbers may have some common factors. Consider the
prime factorizations of 90 and 105 shown below.
90  2  3  3  5
105  3  5  7
The integers 90 and 105 have 3 and 5 as common prime factors. The
product of these prime factors, 3 · 5 or 15, is called the greatest
common factor (GCF) of 90 and 105.
DEFINITION OF GREATEST COMMON FACTOR:
The greatest common factor of two or more integers is the product of
the prime factors common to the integers.
Note: The GCF of two or more monomials is the product of their
common factors, when each monomial is expressed as a product of
prime factors.
Ex. 4: Find the GCF of 54, 63, and 180
54  2  3  3  3
63  3  3  7
180  2  2  3  3  5
Factor each number.
Then circle the common factors.
The GCF of 54, 63, and 180 is 3 · 3 or 32 or 9.
Ex. 5: Find the GCF of 8a2b and 18a2b2c
8a b  2  2  2  a  a  b
2
Factor each number.
18a b c  2  3  3  a  a  b  b  c
2 2
Then circle the common factors.
The GCF of 8a2b and 18a2b2c is 2 · a · a · b or 2a2b.