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Table of Contents
· Prime and Composite Numbers
· Prime Factorization
Whole Numbers
· Common Factors
· Greatest Common Factor
· Relatively Prime
· Least Common Multiple
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Prime and Composite Numbers
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1
The smallest prime number is _______.
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2
49 is not a prime number.
True
False
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3
This list contains 3 prime numbers:
1, 2, 3, 5, 9, and 12
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4
This list contains 3 prime numbers:
5, 9, 20, 31, 42, 53, and 63
True
True
False
False
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5
This list contains 3 prime numbers:
5, 9, 20, 31, 42, 53, and 63
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6
This list contains 3 prime numbers:
15, 19, 23, 37, 47, 55, and 63
True
True
False
False
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7
This list contains 3 prime numbers:
25, 29, 33, 38, 45, 57, and 76
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The Sieve of Erastosenes
True
Find the prime numbers by sifting out the multiples of each
prime.
False
Example:
2 is prime.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14...
How do we know that the multiples of 2 are not prime?
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The Sieve of Erastosenes
A Composite Number can be divided evenly
by numbers other than 1 or itself.
Sift out
the
multiples
of each
prime.
Examples:
What are
you left
with?
1 is NOT composite. Why not?
X
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Is 18 prime or composite?
Explain
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Is 63 prime or composite?
Explain
18 is composite because it can be
divided evenly by more than 1 and
itself. 18 can be evenly divided by:
1, 2, 3, 6, 9, and 18.
63 is composite because it can be
divided evenly by more than 1 and
itself. 63 can be evenly divided by:
1, 3, 7, 9, 21, and 63.
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8
43 is _________
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9
30 is _________
A
Prime
A
Prime
B
Composite
B
Composite
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10 33 is _________
A
Prime
B
Composite
Factoring a Number
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Factors
Factors are the numbers you multiply together to get another number.
Example:
3 and 6 are factors of 18, because 3 x 6 = 18.
Also, 2 x 9 =18, so 2 and 9 are also factors of 18.
What are two other factors of 18?
Prime Factorization
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Process for factoring a number into primes
is the process of factoring a number so that all of the
factors are prime numbers.
What is the prime factorization of 18?
1. Divide the given number by the smallest prime number possible.
2. Continue to divide by the smallest prime number possible.
3. Keep dividing until the quotient (answer) is one.
Example:
12 = 2 x 2 x 3
= 22 x 3
2 18
18 = 2 x 3 x 3
3 9
3 3
1
2 12
2 6
3 3
1
click
for
= 2 x 32
answer
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What is the prime factorization of 24?
2 24
24 = 2 x 2 x 2 x 3
2 12
click
= 2 x 3 for
answer
3
2 6
3 3
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11 What is the prime factorization of 30?
A
2x3x5
B
6x5
C
5x6
D
2 x 15
1
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12 What is the prime factorization of 24?
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13 What is the prime factorization of 45?
A
3x8
A
3 x 15
B
2x2x6
B
32 x 5
C
23 x 3
C
9x 5
D
2x2x2x3
D
52 x 3
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14 What is the prime factorization of 60?
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15 What is the prime factorization of 100?
A
2 x 3 x 10
A
2 x 3 x 10
B
2x5x2x3
B
2x5x2x3
2
C
2 x3x5
C
22 x 3 x 5
D
22 x 15
D
22 x 15
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Common Factors
A common factor is a number that is a factor of two or more numbers.
Find the common factors of 12 and 16.
Factors of 12: 1, 2, 3,
4, 6,
click
for12answer
Factors of 16: 1, 2, 4,click
8, 16
for answer
Common factors: 1, 2, 4
click for answer
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Common Factors
Find the common factors of 18 and 24.
Factors of 18: 1, 2, 3,click
6, 9,for18answer
Factors of 24: 1, 2, 3,click
4, 6,for8,12,
24
answer
Common factors: 1, 2, 3, 4, 6
click for answer
What is the Greatest Common Factor?
What is the Greatest Common Factor?
Greatest Common Factor: 4
Greatest Common Factor: 6
click for answer
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16 The greatest common factor for 12 and 48 is ____.
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17 The greatest common factor for 24 and 36 is ____.
A
2
A
2
B
4
B
4
C
6
C
6
D
12
D
12
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18 The greatest common factor for 42 and 64 is ____.
click for answer
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19 The greatest common factor for 50 and 100 is ____.
A
2
A
5
B
4
B
10
C
6
C
25
D
8
D
50
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20 The greatest common factor for 36 and 90 is ____.
A
3
B
9
C
12
D
18
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Greatest Common Factor
We can use prime factorization to find the greatest common factor
(GCF).
1. Factor the given numbers into primes.
2. Circle the factors that are common.
3. Multiply the common factors together to find the
common factor.
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greatest
21 Use prime factorization to find the GCF of 18 and 44.
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22 Use prime factorization to find the GCF of 28 and 70.
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23 Use prime factorization to find the GCF of 55 and 110.
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24 Use prime factorization to find the GCF of 52 and 78.
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25 Use prime factorization to find the GCF of 72 and 75.
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Relatively Prime:
Two or more numbers are relatively prime if
their greatest common factor is 1.
Example:
15 and 32 are relatively prime because their GCF is 1.
Name two numbers that are relatively prime.
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26 Identify at least two numbers that are relatively prime to 9.
A
16
B
15
C
28
D
36
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27 7 and 35 are not relatively prime.
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28 Name a number that is relatively prime to 20.
True
False
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29 Name a number that is relatively prime to 5 and 18.
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30 Find two numbers that are relatively prime
A
7
B
14
C
15
D
49
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A multiple of a whole number is the product of the number
and any nonzero whole number.
Least Common Multiple
A multiple that is shared by two or more numbers is a
common multiple.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...
Multiples of 14: 14, 28, 42, 56, 70, 84,...
The least of the common multiples of two or more numbers
is the least common multiple (LCM). The LCM of 6 and
14 is 42.
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Find the least common multiple of 18 and 24.
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31 Find the least common multiple of 10 and 14.
Multiples of 18: 18, 36, 54, 72, ...
A
2
Multiples of 24: 24, 48, 72, ...
B
20
LCM: 72
C
70
D
140
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32 Find the least common multiple of 5 and 30.
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33 Find the least common multiple of 9 and 15.
A
6
A
3
B
10
B
30
C
30
C
45
D
150
D
135
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34 Find the least common multiple of 3, 6, and 9.
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35 Find the least common multiple of 16, 20, and 30.
A
3
A
80
B
12
B
100
C
18
C
240
D
36
D
320
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Another way to find the least common multiple (LCM) is
to factor the numbers into primes and then multiply all
of the factors, using each common factor only once.
Find the least common multiple (LCM) by factoring the
number into primes and then multiply all of the factors,
using each common factor only once.
Example: Find the LCM of 12 and 18.
Example: Find the LCM of 16 and 28.
2 12
2 18
2 6
3 9
3 3
1
33
1
12 = 2 x 2 x 3
2 16
18 = 2 x 3 x 3
2 8
LCM: 2 x 3 x 2 x 3
= 36
16 = 2 x 2 x 2 x 2
2 28
2 14
2 4
2 2
7 7
1
28 = 2 x 2 x 7
LCM: 2 x 2 x 2 x 2 x 7
= 112
1
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Find the least common multiple (LCM) by factoring the numbers into
primes and then multiply all of the factors, using each common factor only
once.
36 Use prime factorization to find the LCM of 12 and 20.
Example: Find the LCM of 10, 12, and 20.
5
5
1
10 = 2 x 5
2 20
2 12
2 10
2 6
3 3
1
2 10
12 = 2 x 2 x 3
5 5
20 = 2 x 2 x 5
1
LCM: 2 x 5 x 2 x 3 x 5
= 300
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37 Use prime factorization to find the LCM of 24 and 60.
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38 Use prime factorization to find the LCM of 9, 15, and 18.
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39 Use prime factorization to find the LCM of 16, 24, and 32.
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41 Use prime factorization to find the GCF of 15, 20, 75.
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40 Use prime factorization to find the LCM of 15, 20, 75.