Download Ch. 2 - Northwest ISD Moodle

Chapter 2
Summary
Key Terms
Property
f least common multiple
f prime factorization
(2.1)
f
f
f
f
f
f Associative Property of
(LCM) (2.2)
Multiplication (2.1)
f common factor (2.3)
f greatest common factor
factor tree (2.1)
power (2.1)
(GCF) (2.3)
base (2.1)
f relatively prime numbers
exponent (2.1)
f Fundamental Theorem
(2.3)
common multiple (2.2)
Theorem
of Arithmetic (2.1)
Determining the Prime Factorization of a Number
The Fundamental Theorem of Arithmetic states that every natural number is either prime
or can be uniquely written as a product of primes. This product of primes is the prime
factorization of the number. A factor tree is a useful tool for organizing the factors of a
number. Powers are often used to express repeated factors.
Example
The prime factorization of 96 using a factor tree is shown. Below the factor tree, the prime
factorization of 96 using powers is shown.
96
8
2
12
2
© Carnegie Learning
3
4
2
4
2
96 5 2 3 2 3 2 3 2 3 2 3 3
5 25 ? 3
2
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Chapter 2
Summary
•
89
Determining the Least Common Multiple
The least common multiple, or LCM, is the least multiple (other than zero) that two or more
numbers have in common.
Example
The least common multiple of 20 and 45 using prime factorization is shown.
20 5 2 3 2 3 5 5 22 ? 5
45 5 3 3 3 3 5 5 32 ? 5
LCM = 22 ? 32 ? 5 5 180
The LCM of 20 and 45 is 180.
Determining the Greatest Common Factor
The greatest common factor, or GCF, is the greatest factor that two or more numbers have
in common.
Example
The greatest common factor of 64 and 120 using prime factorization is shown.
64 5 26
120 5 23 ? 3 ? 5
GCF 5 23
58
© Carnegie Learning
The GCF of 64 and 120 is 8.
90
•
Chapter 2
Prime Factorization and the Fundamental Theorem of Arithmetic
Using GCF and LCM to Solve Problems
Common factors help determine how to divide or share things equally. Common multiples
help determine how things with different cycles can occur at the same time.
Example
a. A florist has 24 daisies, 40 zinnias, and 32 snapdragons. She wants to divide the
flowers evenly to make bouquets for her display case. The florist can determine the
greatest common factor to calculate how many bouquets he can create.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Factors of 32: 1, 2, 4, 8, 16, 32
The GCF is 8.
The florist can make eight bouquets because the greatest common factor of 24, 40, and
32 is 8.
b. Carl has gym class every 4 school days and music class every 3 school days. If Carl
has both gym class and music class today, in how many school days will he have both
classes on the same day again?
Multiples of 4: 4, 8, 12, . . .
Multiples of 3: 3, 6, 9, 12, . . .
The LCM is 12.
© Carnegie Learning
Carl will have both classes on the same day in 12 school days.
Chapter 2
Summary
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91