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Factors and Prime Factorization
Goal: Write the prime factorization of numbers.
Vocabulary
Prime number:
A whole number greater than 1 whose only
positive factors are 1 and itself.
Composite number:
A whole number greater than 1 that has
positive factors other than 1 and itself.
Prime factorization:
Writing a number as the product of prime
numbers.
A diagram that can be used to write the prime
factorization of a number.
Factor tree:
A number, a variable, or a product of a number and
one or more variables.
Monomial:
EXAMPLE
1
Writing Factors
Band A band with 48 members is arranged into a rectangular formation with
the same number of members in each row. There should be at least 2 rows
with at least 3 members in each row. Find the number of possible formations
by finding the factors of 48.
Solution
1. Write 48 as a product of two numbers in all possible ways.
1 48
2 24
3 16
4 12
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The factors of 48 are 1, 2 , 3, 4, 6 , 8 , 12, 16 , 24, and 48 .
2. Use these factors to find all the possible formations.
Rows
Members
Rows
Members
2 24
68
3 16
86
4 12
12 4
Rows
Members
16 3
ANSWER There are 7 possible rectangular formations.
Lesson 4.1
Factors and Prime Factorization
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67
Write all the factors of the number.
Guided Practice
1. 30
You can use
divisibility tests
to help find all
the factors of a
composite number.
2. 31
1, 2, 3, 5, 6,
10, 15, 30
EXAMPLE
2
3. 45
1, 31
4. 51
1, 3, 5, 9,
15, 45
1, 3, 17, 51
Identifying Prime and Composite Numbers
Write all the factors of the number and tell whether it is prime or composite.
Number
Factors
Prime or Composite?
1, 2, 3 , 6, 9 , 18
a. 18
Composite
1, 7, 49
b. 49
Composite
c. 58
1, 2 , 29 , 58
Composite
d. 67
1, 67
Prime
1, 131
e. 131
f.
Prime
1, 3, 5 , 11 , 15, 33 , 55, 165
165
EXAMPLE
3
Composite
Writing Prime Factorization
Write the prime factorization of 540.
Two factor trees are shown. Notice that each factor tree produces the same
prime factorization, differing only in the order of the factors.
540
540
Notice that more
than one factor tree can
be used to write the
prime factorization of
some numbers. Another
factor tree for the prime
factorization of 540
begins with 18 p 30.
p
27
9
p
3 p
3
3
p 3
20
p 4 p
5
p
p
12
5
5
p 2 p 2 p 5
5
108
p
9
5
p
3
p
4
p
3
p
3
p
3
p
2
p
2
p
3
p
So, 540 2 p 2 p 3 p 3 p 3 p 5 .
ANSWER Using exponents, the prime factorization of 540 is 22 p 33 p 5 .
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Chapter 4
Notetaking Guide
3
Guided Practice
Tell whether the number is prime or composite. If
it is composite, write its prime factorization using exponents.
5. 28
6. 61
composite;
22 p 7
4
EXAMPLE
7. 72
prime
8. 600
composite;
23 p 32
composite;
23 p 3 p 52
Factoring a Monomial
Factor the monomial 24xy3.
24xy 3 2 p 2 p 2 p 3
p x p y3
Factor 24.
2p2p2p3pxpypypy
Guided Practice
Write y 3 as y p y p y .
Factor the monomial.
10. 20t 3
9. 5ab
5papb
11. 36x 2y 4
2p2p3p3pxpxpypypypy
2p2p5ptptpt
12. 56p3q
2p2p2p7pppppppq
Lesson 4.1
Factors and Prime Factorization
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