Topic 1: Prime Factorization
for use after
Bits and Pieces I
Investigation 1
Any composite number can be written as a product using its factors. The
of a number is the product of prime factors equal to
that number.
You can use factor trees to find the prime factorization of a number. For
example, to make a factor tree for 28, begin by choosing two numbers
whose product is 28. Continue dividing each number into two factors until
each of the branches ends in a prime number. Below is a sample factor tree
for 28.
The prime factorization of 28 is 2 3 2 3 7, or 2
2
3 7.
The small raised number in 2
2
3 7 is an exponent. An
tells
how many times a factor is multiplied repeatedly. For example, the
expression 3
2
3 5
3
means 3 3 3 3 5 3 5 3 5.
A. 1. List all of the factors of 90. Which factors are prime?
2. Which factor pairs could you use to start a factor tree for 90?
3. Make two different factor trees for 90. What do you notice about
the prime factorization of each tree?
4. How can you write the prime factorization of 90 using exponents?
B. 1. Are each of the prime factors of a number included in the prime
factorization at least once?
2. Does it matter which two factors you choose for the first line of a
factor tree?
exponent
7
?
4
7
? 2 ? 2
28
prime factorization
Exercises
For Exercises 1–9, find the prime factorization of each number.
1. 20
2. 36
3. 50
4. 85
5. 100
6. 189
7. 525
8. 639
9. 1,000
10. Use exponents to write the prime factorizations you found in
Exercises 1–9.
For Exercises 11–12, copy and complete the factor trees.
11.
12.
For Exercises 13–16, consider the prime factorization of each of the
whole numbers from 2 through 50.
13. Which of the whole numbers from 2 to 25 are prime?
14. Which of the whole numbers from 20 to 30 are the product of exactly
three different prime factors?
15. Which of the whole numbers from 30 to 40 have 5 as a prime factor?
16. Which of the whole numbers from 40 to 50 have a prime factorization
with only odd prime factors?
17. a. Make three different factor trees for 360.
b. How many different pairs of numbers could you use to start a factor
tree for 360?
760
20
■
?
2
■
?
?
2
■
?
■
■■
?
5
?
? 5
?
■
2
?
2
?
2
■
2
?
2
2
■
?
?
?
27
■
?
2 ? 2
■
33
? 3
?
22
?
?
2 ? 3 ?
?
216
Topic 1: Prime Factorization
Teaching Guide
Mathematical Goals
•
Use a factor tree to find a prime factorization
•
Write a prime factorization using exponents
Vocabulary
•
prime factorization
•
exponent
At a Glance
Before beginning Topic 1, review the concepts of prime numbers and
factors with students. You can also review the concept of exponents with
students. Explain that an expression in the form
a
b
is called a power;
a
is
the base, and
b
is the exponent.
Before Problem 1.1, ask:
•
What is the definition of a prime number?
•
Is 1 considered a prime number?
•
How can you find the factors of a number?
•
How can you use exponents to rewrite the expression 5
3
5
3
5?
the expression 2
3
2
3
5
3
5
3
5?
During Problem 1.1, you may want to review divisibility rules to help the
students find factors. Ask:
•
How do you know if a number is divisible by 2? by 3? by 5?
•
How do you know if an number is divisible by 6? by 10?
After Problem 1.1B, explain to students that the prime factorization of a
number is unique. Students may be unsure about the order in which they
should write the numbers in a prime factorization. Explain that the prime
factors are usually written with primes listed in increasing order. Point out
to students that the order of the prime factors does not change the prime
factorization, because multiplication is commutative. Remind students to
include repeated prime factors in a prime factorization, or to use exponent
notation.
Homework Check
When reviewing Exercise 17, ask:
•
Is there a factor pair that cannot be used to start a factor tree?
•
Does the factor pair you choose to start a factor tree matter? Will
you always get the same result?
PACING
1 day
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Assignment Guide for Topic 1
Core
1–17
Answers to Topic 1
Problem 1.1
A. 1.
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30,
45, 90
Prime factors of 90: 2, 3, 5
2.
2 and 45, 3 and 30, 5 and 18, 6 and 15, 9
and 10
3.
Answers may vary. Sample:
4.
2 3 3
2
3 5
B. 1.
yes
2.
no
Exercises
1.
2 3 2 3 5
2.
2 3 2 3 3 3 3
3.
2 3 5 3 5
4.
5 3 17
5.
2 3 2 3 5 3 5
6.
3 3 3 3 3 3 7
7.
3 3 5 3 5 3 7
8.
3 3 3 3 71
9.
2 3 2 3 2 3 5 3 5 3 5
10.
2
2
3 5, 2
2
3 3
2
, 2 3 5
2
, 5 3 17,2
2
3 5
2
,
3
3
3 7,3 3 5
2
3 7,3
2
3 71,2
3
3 5
3
11.
12.
13.
2, 3, 5, 7, 11, 13, 17, 19, 23
14.
21, 22, 26
15.
30, 35, 40
16.
41, 43, 45, 47, 49
17. a.
Answers may vary. Sample:
b.
11
360
15
24
3
3
3
?
?
??
?
??
5
5
5
?
2
? ?
24
46
22
?
3
360
2
?
180
23
??
60
23
? ?
2
?
30
23
? ?
2
?
5
?
6
23
? ?
2
?
5
?
2
?
3
360
36
10
6
?
?
6
?
5 2
3
?
2
?
3
?
2
?
5
?
2
760
20
38
4
?
?
5
??
2 19
2
?
2
?
52
??
19
216
2
?
108
22
??
54
22
? ?
2
?
27
22
? ?
2
?
3
?
9
22
? ?
2
?
3
?
3
?
3
90
6
?
15
2
?
3
?
3
?
5
90
2
?
45
25
??
9
25
? ?
3
?
3
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