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FACTORS AND
GREATEST COMMON FACTORS
A PRIME NUMBER is a whole number, greater than 1,
whose only factors are 1 and itself.
A COMPOSITE NUMBER is a whole number, greater
than 1, that has more than two factors are 1 and itself.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
PRIME NUMBERS ARE GREATER THAN 1.
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
So, 2 is the smallest prime number.
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Any number divisible by 2 (evens) are not prime.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
So, 3 is the next prime number.
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Any number divisible by 3, is not a prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
The next prime number is 5.
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Any number divisible by 5 is not a prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
7 is the next prime number.
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Any number divisible by 7 is not a prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
11 is the next prime number.
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Numbers divisible by 11 are already crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
13 is the next prime number.
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Numbers divisible by 13 are already crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Next prime number is 17. All multiples crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
19 is the next prime number. All multiples crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The remaining numbers have no multiples uncrossed.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The numbers circled are prime numbers.
Except for 1, all the rest are composite numbers.
PRIME FACTORIZATION
Write the prime factorization of 80.
FACTOR TREE
80
10
2
8
5 2
4
2
24•5
2
PRIME FACTORIZATION
Write the prime factorization of 80.
FACTOR TREE
Start with smallest
prime number
80
10
2
4
2
2
80
40
20
2
10
5
5
1
2
2
8
5 2
24•5
INVERTED DIVISION
2
24•5
PRIME FACTORIZATION
OF A MONOMIAL
Factor -36x2y3z completely:
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
Factor 54x4yz3 completely:
2•3•3•3•x•x•x•x•y•z•z•z
PRIME FACTORIZATION
OF A MONOMIAL
Find the GCF for -36x2y3z and 54x4yz3
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
2•3•3•3•x•x•x•x•y•z•z•z
PRIME FACTORIZATION
OF A MONOMIAL
Find the GCF for -36x2y3z and 54x4yz3
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
2•3•3•3•x•x•x•x•y•z•z•z
CIRCLE THE COMMON FACTORS
PRIME FACTORIZATION
OF A MONOMIAL
Find the GCF for -36x2y3z and 54x4yz3
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
2•3•3•3•x•x•x•x•y•z•z•z
GCF = 2 • 3 • 3 • x • x • y • z = 18x2yz
MULTIPLY THE COMMON FACTORS
PRIME FACTORIZATION
OF A MONOMIAL
Find the GCF for 28m3n and 21m2n5
2•2•7•m•m•m•n
3•7•m•m•n•n•n•n•n
GCF = 7 • m • m • n = 7m2n
MULTIPLY THE COMMON FACTORS
FLASH CARDS
WHAT IS THE GCF?
20 and 30
10
FLASH CARDS
WHAT IS THE GCF?
4x and 6y
2
FLASH CARDS
WHAT IS THE GCF?
6m and 12m
6m
FLASH CARDS
WHAT IS THE GCF?
8xy and 12xz
4x
FLASH CARDS
WHAT IS THE GCF?
2
10a b
and
2ab
2
14ab
FACTORING USING THE
DISTRIBUTIVE PROPERTY
Recall the Distributive Property:
Example 1: 5(x + y) = 5x + 5y
Example 2: 2x(x + 3) = 2x2 + 6x
In this section, you will be learning
how to use the Distributive Property
backwards…..or FACTORING.
In other words, start with  5x + 5y
and factor it into  5(x + y)
FACTORING USING THE
DISTRIBUTIVE PROPERTY
In an algebraic expression, the quantities
being multiplied are called FACTORS.
EXAMPLES
10  the factors are 2 and 5.
2xy  the factors are 2, x and y.
5(x + y)  the factors are 5 and (x + y)
3x(x + 7)  the factors are 3, x and (x + 7)
FACTORING USING THE
DISTRIBUTIVE PROPERTY
If we take a look at two expressions:
3x
and
5x
x is a factor in common to both
x is a monomial
So, x is a
Common Monomial Factor
of 3x and 5x.
CMF
FACTORING USING THE
DISTRIBUTIVE PROPERTY
Let’s factor 4x + 8y
What is the CMF (or GCF)
for the two terms?
Answer: 4
Write down the 4 followed by (
4(
Then ask, “what times 4 = 4x”?
Answer: x
Write down the x after the (
4(x
Then ask, what times 4 = 8y?
Answer: 2y
Add that to the “4(x” and close the parentheses.
Final Answer: 4(x + 2y)
FACTORING PRACTICE
Factor: 3m + 12
Step 1: What is the CMF? 3
Step 2: 3 times ? = 3m
3(m
Step 3: 3 times ? = 12 3(m+ 4)
3m + 12 = 3(m + 4)
FACTORING PRACTICE
Factor: m2 – 8m
Step 1: What is the CMF?
Step 2: m times ? = m2
m
m(m
Step 3: m times ? = -8m
m(m – 8)
m2 – 8m = m(m – 8)
FACTORING PRACTICE
Factor: 10x2y – 5xy + 15y
5y
Step 1: What is the CMF?
5y(2x2
Step 2: 5y times ? = 10x2y
Step 3: 5y times ? = – 5xy 5y(2x2 – x
Step 4: 5y times ? = + 15y
5y(2x2 – x + 3)
10x2y + 5xy + 15y
= 5y(2x2 – x + 3)
TRY THESE
1. Factor 2x – 12
2(x – 6)
2. Factor 12ab + 8bc
4b(3a + 2c)
3. Factor 6x2y – 3x3y2 + 5x4y3
x2y(6 – 3xy + 5x2y2)
FLASH CARDS
WHAT IS THE CMF?
6x + 15
3
FLASH CARDS
WHAT IS THE CMF?
12m2 – 8m
4m
FLASH CARDS
WHAT IS THE CMF?
3a2 – 7b2
1
so, the
expression is a
prime polynomial.
FLASH CARDS
WHAT IS THE CMF?
– 4b3 + 8b2c – 6bc2
2b
FLASH CARDS
WHAT IS THE CMF?
a3b+ a2b2 – ab3
ab
FLASH CARDS
FILL IN THE BLANK?
ab(___) = 3ab2
3b
FLASH CARDS
FILL IN THE BLANK?
3m(___) = 6m2
2m
FLASH CARDS
FILL IN THE BLANK?
5xy(___) = 15x2y
3x
FLASH CARDS
FILL IN THE BLANK?
2cd(___) = –12c2d3
2
–6cd
ONE MORE ITEM
TO SOLVE EQUATIONS IN THIS SECTION YOU WILL USE
THE ZERO PRODUCT PROPERTY
For any real numbers a and b,
if ab = 0, then either a = 0 or b= 0.
SOLVING EQUATIONS
STEP 1: Set equation equal to zero
STEP 2: Factor the left side of the equation
STEP 3: Set each factor equal to zero
STEP 4: Solve each equation
SOLVING EQUATIONS
Solve: 3m2 + 12m = 3m
3m2 + 12m = 3m
STEP 1: Set equation = 0
-3m
3m2 + 9m = 0
3m(m + 3) = 0
STEP 2: Factor left side
STEP 3: Set each factor = 0
3m = 0 or m + 3 = 0
3
STEP 4: Solve each equation
-3m
3
-3
m = 0 or m = -3
-3
SOLVING EQUATIONS
Solve: 6x2 = -8x
6x2 = -8x
STEP 1: Set equation = 0
+8x
6x2 + 8x = 0
2x(3x + 4) = 0
STEP 2: Factor left side
STEP 3: Set each factor = 0
2x = 0 or 3x + 4 = 0
2
STEP 4: Solve each equation
+8x
2
-4
-4
3x = -4
3
x = 0 or x =
3
4

3