Download Alg - 1- 8.1

8-1 Factors and Greatest Common Factors
Warm Up
Tell whether the second number is a factor
of the first number
1. 50, 6
no
2. 105, 7
3. List the factors of 28.
±14, ±28
yes
±1, ±2, ±4, ±7,
Tell whether each number is prime or
composite. If the number is composite, write
it as the product of two numbers.
4. 11 prime
Holt Algebra 1
5. 98 composite; 49  2
8-1 Factors and Greatest Common Factors
Simplify.
A. 4–3
B. 70
C. (–5)–4
D. –5–4
Holt Algebra 1
1
1
 3
4
64
1
1
1


4
 5 625
1
1
 4 
5
625
8-1 Factors and Greatest Common Factors
Learning Target
Students will be able to:Write the
prime factorization of numbers and find
the GCF of monomials.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
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

Holt Algebra 1






8-1 Factors and Greatest Common Factors
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

Holt Algebra 1






8-1 Factors and Greatest Common Factors
Remember!
A prime number has exactly two factors, itself
and 1. The number 1 is not prime because it only
has one factor.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Write the prime factorization of 98.
2 98
7 49
7
98 = 2  72
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Write the prime factorization of each number.
a. 40
2 40
2 20
2 10
5
40  2  5
3
Holt Algebra 1
b. 33
33  3 11
3 33
11
8-1 Factors and Greatest Common Factors
Write the prime factorization of each number.
c. 49
d. 19
7 49
7
49  7
Holt Algebra 1
2
19
8-1 Factors and Greatest Common Factors
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 factors is 4.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of numbers.
100 and 60
2 100
2 50
5 25
5
2 60
2 30
3 15
5
100  2  2  5  5
60  2  2  3  5
The GCF of 100 and 60 is 20.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of numbers.
26 and 52
2 26
13
2 52
2 26
13
26  2 13
52  2  2 13
The GCF of 26 and 52 is 26.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of numbers.
15 and 25
3 15
5
5 25
5
15  3  5
25  5  5
The GCF of 15 and 25 is 5.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of monomials.
15x3 and 9x2
3 15
5
3 9
3
15 x  3  5  x  x  x
2
9x  3 3 x  x
3
3
2
2
The GCF of 15 x and 9 x is 3x .
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of monomials.
8x2 and 7y3
2 8
2 4
2
1 7
8x  2  2  2  x  x
3
7y  7 y y y
2
2
3
The GCF of 8x and 7 y is 1.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Helpful Hint
If two terms contain the same variable raised to
different powers, the GCF will contain that
variable raised to the lower power.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of monomials.
18g2 and 27g3
2 18
3 9
3
3 27
3 9
3
18g  2  3  3  g  g
3
27 g  3  3  3  g  g  g
2
2
3
2
The GCF of 18g and 27 g is 9 g .
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Find the GCF of each pair of monomials.
16a6 and 9b
2 16
3 9
2 8
3
2 4
2
6
16a  2  2  2  2  a  a  a  a  a  a
9b  3  3  b
2
The GCF of 16a and 9b is 1.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
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?
The 18 chocolate and 24 regular milk cartons must
be divided into groups of equal size. The number of
cartons in each row must be a common factor of 18
and 24.
2 18 2 24 The GCF of 18 and 24 is 6.
18  2  3  3 3 93 22 126
24  2  2  2  3
Holt Algebra 1
3
8-1 Factors and Greatest Common Factors
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.
HW pp. 527-529/17-46,56-59, 60-68 Even, 69-75
Holt Algebra 1