factors and greatest common factors a prime number is a whole number, greater than 1, whose only...
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FACTORS ANDGREATEST 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
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
PRIME NUMBERS ARE GREATER THAN 1.
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
So, 2 is the smallest prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Any number divisible by 2 (evens) are not prime.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
So, 3 is the next prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Any number divisible by 3, is not a prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
The next prime number is 5.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Any number divisible by 5 is not a prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
7 is the next prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Any number divisible by 7 is not a prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
11 is the next prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Numbers divisible by 11 are already crossed out.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
13 is the next prime number.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Numbers divisible by 13 are already crossed out.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Next prime number is 17. All multiples crossed out.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
19 is the next prime number. All multiples crossed out.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
The remaining numbers have no multiples uncrossed.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
ERATOSTHENE’S SIEVE
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
The numbers circled are prime numbers.Except for 1, all the rest are composite numbers.
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
PRIME FACTORIZATION
FACTOR TREE
Write the prime factorization of 80.
80
10 8
2 4
2 2
24•5
2 5
PRIME FACTORIZATION
FACTOR TREE INVERTED DIVISION
Write the prime factorization of 80.
802402
2
2 20
10
551
24•5
Start with smallestprime numberStart with smallestprime number
FACTOR TREE
80
10 8
2 4
2 2
24•5
2 5
PRIME FACTORIZATIONOF 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 FACTORIZATIONOF A MONOMIAL
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
Find the GCF for -36x2y3z and 54x4yz3
2 • 3 • 3 • 3 • x • x • x • x • y • z • z • z
PRIME FACTORIZATIONOF A MONOMIAL
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
Find the GCF for -36x2y3z and 54x4yz3
2 • 3 • 3 • 3 • x • x • x • x • y • z • z • z
CIRCLE THE COMMON FACTORS
PRIME FACTORIZATIONOF A MONOMIAL
-1 • 2 • 2 • 3 • 3 • x • x • y • y • y • z
Find the GCF for -36x2y3z and 54x4yz3
2 • 3 • 3 • 3 • x • x • x • x • y • z • z • z
MULTIPLY THE COMMON FACTORS
GCF = 2 • 3 • 3 • x • x • y • z = 18x2yz
PRIME FACTORIZATIONOF A MONOMIAL
2 • 2 • 7 • m • m • m • n
Find the GCF for 28m3n and 21m2n5
3 • 7 • m • m • n • n • n • n • n
MULTIPLY THE COMMON FACTORS
GCF = 7 • m • m • n = 7m2n
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?
10a2b and 14ab2
2ab
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 + 5yand factor it into 5(x + y)
FACTORING USING THE DISTRIBUTIVE PROPERTY
In an algebraic expression, the quantities being multiplied are called FACTORS.
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)
EXAMPLES
10 the factors are 2 and 5.
FACTORING USING THE DISTRIBUTIVE PROPERTY
If we take a look at two expressions:
and 5xx is a factor in common to both
3x
x is a monomial
So, x is a Common Monomial Factor
of 3x and 5x.Common Monomial Factor
CMFCMF
FACTORING USING THE DISTRIBUTIVE PROPERTY
Let’s factor 4x + 8yWhat 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? m
Step 2: m times ? = m2 m(m
Step 3: m times ? = -8mm(m – 8)
m2 – 8m = m(m – 8)
FACTORING PRACTICE
Factor: 10x2y – 5xy + 15y
Step 1: What is the CMF? 5yStep 2: 5y times ? = 10x2y 5y(2x2
Step 3: 5y times ? = – 5xy 5y(2x2 – x
10x2y + 5xy + 15y = 5y(2x2 – x + 3)
Step 4: 5y times ? = + 15y5y(2x2 – x + 3)
TRY THESE
1. Factor 2x – 12
2. Factor 12ab + 8bc
3. Factor 6x2y – 3x3y2 + 5x4y3
2(x – 6)
4b(3a + 2c)
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
1so, 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
–6cd2
ONE MORE ITEM
TO SOLVE EQUATIONS IN THIS SECTION YOU WILL USETHE 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
STEP 1: Set equation = 0
STEP 2: Factor left side
STEP 3: Set each factor = 0
STEP 4: Solve each equation
Solve: 3m2 + 12m = 3m
3m2 + 12m = 3m-3m -3m
3m2 + 9m = 03m(m + 3) = 0
3m = 0 or m + 3 = 03 3 -3 -3
m = 0 or m = -3
SOLVING EQUATIONS
STEP 1: Set equation = 0
STEP 2: Factor left side
STEP 3: Set each factor = 0
STEP 4: Solve each equation
Solve: 6x2 = -8x
6x2 = -8x+8x +8x
6x2 + 8x = 02x(3x + 4) = 0
2x = 0 or 3x + 4 = 02 2 -4 -4
3x = -43 3
x = 0 or x =34