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