Download 1.2 Prime Factorization

1.2 Prime Factorization
GOAL
Express a composite number as the product of prime factors.
Learn about the Math
Teo and Sheree need passwords for their e-mail accounts. They want to
use numbers that they can remember, but will be difficult for others to
figure out. Sheree uses the prime factorization of her address, 1050, to
form part of her password. Teo uses the prime factorization of his date of
birth, 330 (March 30).
4
5
1050
11
18
25
1
6
12
19
26
2
7
13
20
27
3
8
9
14
15
21
28
22
29
16
23
30
prime factorization
the representation of a
composite number as
the product of its
prime factors; for
example, the prime
factorization of 24 is
24 2 2 2 3, or
23 3; usually, the
prime numbers are
written in order from
least to greatest
10
17
24
31
? What are Teo’s and Sheree’s passwords?
Example 1: Using a factor tree to determine prime factors
If Sheree’s address is 1050, what will her password be?
Sheree’s Solution
1050
10
5
105
2
5
21
7
3
1050 2 3 5 5 7
2 3 52 7
My e-mail password will be my
address followed by its prime factors:
105023557.
8
Chapter 1
I know that 10 is a factor of 1050. I divided by
10 to determine another factor, 105. I used a
factor tree to show these two factors.
5 is a factor of both 10 and 105. I divided each
number by 5 to determine two more factors,
2 and 21.
I continued to calculate factors until only prime
factors were left at the ends of the branches.
I wrote the prime numbers in order, from least to
greatest, to create the prime factorization of 1050.
Then I used exponents to write the prime
factorization a different way.
NEL
Example 2: Using repeated division to determine prime factors
If Teo’s date of birth is 330, what will his password be?
Teo’s Solution
165
23
3
0
33
51
6
5
23
3
0
11
33
3
51
6
5
23
3
0
330 2 3 5 11
My e-mail password will be my date of
birth followed by its prime factors:
33023511.
I divided by prime factors.
The prime number 2 is a factor of 330. So, I
divided by 2 to determine another factor, 165.
The prime number 5 is a factor of 165. I divided
by 5 to determine another factor, 33.
I divided by 3 to get the last prime factor.
I wrote the prime factors as a multiplication
sentence to show the prime factorization of 330.
Reflecting
1. What might Sheree’s factor tree have looked like if she had
started dividing by 50 instead of 10?
2. Does the order in which you divide by factors change
the prime factorization? Provide an example to support
your answer.
3. a) What divisibility rules might Sheree have used
to determine some factors of 1050, without
dividing the number?
b) What divisibility rules might Teo have used
to determine some factors of 330, without
dividing the number?
4. If you used Sheree’s strategy and Teo’s strategy
for the same number, would you get the same
prime factorization? Explain. Use an
example to support your explanation.
NEL
Number Relationships
9
Work with the Math
Example 3: Writing the prime factorization of a composite number
What is the prime factorization of 1470?
Solution A: Creating a factor tree
Solution B: Using repeated division
1470
10
5
147
2
3
49
7
7
Write the prime numbers at the ends of the
branches in order, from least to greatest, to
show the prime factorization.
1470 2 3 5 7 7
2 3 5 72
A
3
72
1
71
4
7
57
3
5
21
4
7
0
Write the prime numbers in the divisors and
quotient in order, from least to greatest, to
show the prime factorization.
1470 2 3 5 7 7
2 3 5 72
8. Factor trees are being used to determine the
prime factorizations of 1755 and 2180.
Checking
5. Determine the prime factorization of each
number.
a) 117
b) 147
c) 220
d) 270
6. a) Rivka used the last four digits of her
telephone number, 1048, followed by
its prime factorization to create her
e-mail password. Determine her
password.
b) Identify any divisibility rules you used.
B
Practising
7. Determine the prime factorization of each
number.
a)
b)
c)
d)
10
100
102
320
375
Chapter 1
e)
f)
g)
h)
412
2055
512
3675
1755
2180
3
585
10
218
a) Explain how you know that each factor
tree is not complete.
b) Copy and complete each factor tree to
determine the prime factorization.
9. Determine the missing number in each
prime factorization.
a)
b)
c)
d)
200 2 2 ■ 5 5
216 23 3■
8281 7 7 13 ■
1568 ■ 5 72
10. a) Determine the prime factorization of
each number.
64
256
1024
b) What does the prime factorization tell
you about each number?
NEL
11. a) Determine the prime factorization of a
three-digit or four-digit composite
number of your choice.
b) Create an e-mail password using the
composite number you chose, followed
by its prime factorization.
12. Why would you determine the prime
factorization of only composite numbers?
Use an example to support your
explanation.
13. How can you use the prime factorization of
a number to determine whether the number
is even or odd? Use an example to support
your explanation.
C
Extending
14. The prime factorization of a number is
23 5 7.
a) Explain how you can use the prime
factorization to determine whether
35 is a factor of the number.
b) Explain how you can use the prime
factorization to determine whether 8 is
a factor of the number.
c) How do you know, without multiplying
the prime factors, that the last digit of
the number is 0?
15. The prime factorizations of two numbers
are shown.
25 3 52 710
38 5 7 4112
a) How do you know that 2 is not a
common factor of the numbers?
b) How do you know that 35 is a common
factor?
c) List another common factor of the
numbers. Explain your reasoning.
NEL
16. a) Multiply any three-digit number by
1001 to get a six-digit number.
1001
■■■
————
b) Divide your six-digit number by 7, 11,
and then 13. What is the quotient?
c) Show the prime factorization of 1001.
d) Explain how you could use the prime
factorization of 1001 to predict the
quotient you calculated in part (b).
17. a) Multiply two different prime numbers.
List all the possible factors of the
product.
b) Repeat part (a) with two other prime
numbers.
c) What do you notice about the number
of factors each product has?
18. a) Multiply three different prime
numbers. List all the possible factors
of the product.
b) Repeat part (a) with three other prime
numbers.
c) What do you notice about the number
of factors each product has?
19. Use your results in questions 17 and 18 to
predict the number of factors the product of
four different prime numbers will have.
Use an example to check your prediction.
20. The prime factorization of a number is
25 38 57 74 11 13. Which
statements are true about the number?
Explain your reasoning.
a)
b)
c)
d)
e)
The number is even.
The number is a multiple of 10.
15 is a factor of the number.
17 is not a factor of the number.
77 is a factor of the number.
Number Relationships
11