Download 7-1 prime factorization and gcf

CHAPTER 7
7-1 FACTORS AND COMMON FACTORS
OBJECTIVES
• Write the prime factorization of numbers.
• Find the GCF of monomials.
FACTORIZATION
• The whole numbers that are multiplied to find a
product are called factors of that product. A
number is divisible by its factors.
• You can use the factors of a number to write the
number as a product. The number 12 can be
factored several ways.
• Factorizations of 12
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PRIME FACTORIZATION
• The order of factors does not change the product,
but there is only one example below that cannot
be factored further. The circled factorization is the
prime factorization because all the factors are
prime numbers. The prime factors can be written in
any order, and except for changes in the order,
there is only one way to write the prime
factorization of a number.
• Factorizations of 12
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REMEMBER
Remember!
A prime number has exactly two factors, itself
and 1. The number 1 is not prime because it only
has one factor.
EXAMPLE#1
• Write the prime factorization of 98.
Method 1 Factor tree
Choose any two factors
of 98 to begin. Keep finding
factors until each branch
ends in a prime factor.
98
2  49
7  7
98 = 2  7  7
SOLUTION
• Method 2 Ladder diagram
Choose a prime factor of 98
to begin. Keep dividing by
prime factors until the
quotient is 1.
2 98
7 49
7 7
1
98 = 2  7  7
The prime factorization of 98 is 2  7  7 or 2  72.
CHECK IT OUT!
• Write the prime factorization of each number.
a. 40
c. 49
The prime factorization of 40 is 2  2  2
 5 or 23  5.
The prime factorization of 49 is 7  7 or
72.
b. 33
d. 19
The prime factorization of 33 is 3  11.
The prime factorization of 19 is 1  19.
GREATEST COMMON FACTOR
• Factors that are shared by two or more whole
numbers are called common factors. The greatest
of these common factors is called the greatest
common factor, or GCF.
• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 32: 1, 2, 4, 8, 16, 32
• Common factors: 1, 2, 4
The greatest of the common factor of
12 and 32 is 4.
EXAMPLE 2A: FINDING THE GCF OF
NUMBERS
Find the GCF of each pair of numbers.
100 and 60
Method 1 List the factors.
factors of 100: 1, 2, 4,
5, 10, 20, 25, 50, 100
• factors of 60: 1, 2, 3, 4, 5, Circle the GCF.
6, 10, 12, 15, 20, 30, 60
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The GCF of 100 and 60 is 20.
EXAMPLE 2B: FINDING THE GCF OF
NUMBERS
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Find the GCF of each pair of numbers.
26 and 52
Method 2 Prime factorization
26 =
2  13
52 = 2  2  13
2  13 = 26
The GCF of 26 and 52 is 26.
CHECK IT OUT!
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Find the GCF of each pair of numbers
A) 12 and 16
Ans: The GCF of 12 and 16 is 4
B) 15 and 25
Ans:The GCF of 15 and 25 is 5.
GCF IN MONOMIALS
• You can also find the GCF of monomials that
include variables. To find the GCF of monomials,
write the prime factorization of each coefficient
and write all powers of variables as products. Then
find the product of the common factors.
EXAMPLE 3A: FINDING THE GCF OF
MONOMIALS
• Find the GCF of each pair of monomials.
• 15x3 and 9x2
• Sol:
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
3
x  x = 3x2
The GCF of 15x3 and 9x2 is 3x2.
EXAMPLE
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Find the GCF of each pair of monomials.
8x2 and 7y3
Sol:
8x2 = 2  2  2 
xx
7y3 =
7
yyy
The GCF 8x2 and 7y3 is 1. There are no common
factors other than 1.
CHECK IT OUT!!
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Find the GCF of each pair of monomials.
A) 18g2 and 27g3
Answer: The GCF of 18g2 and 27g3 is 9g2.
B) 16a6 and 9b
Answer: GCF is 1
APPLICATION
• A cafeteria has 18 chocolate-milk cartons and 24
regular-milk cartons. The cook wants to arrange the
cartons with the same number of cartons in each
row. Chocolate and regular milk will not be in the
same row. How many rows will there be if the cook
puts the greatest possible number of cartons in
each row?
SOLUTION
• The greatest possible number of milk cartons in
each row is 6. Find the number of rows of each type
of milk when the cook puts the greatest number of
cartons in each row.
18 chocolate milk cartons
= 3 rows
6 containers per row
24 regular milk cartons
6 containers per row
= 4 rows
When the greatest possible number of
types of milk is in each row, there are 7
rows in total.
STUDENT GUIDED PRACTICE
• Do even problems from 2-15 in your book page 459
HOMEWORK
• Do even problems 17-32 in your book page 459
CLOSURE
• Today we learned about prime factorization and
the greatest common factor
• Next class we are going to keep learning about
factorization