GREATEST COMMON FACTORS 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.

PRIME NUMBERS ARE GREATER THAN 1. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 PRIME NUMBERS ARE GREATER THAN 1.

So, 2 is the smallest prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 So, 2 is the smallest prime number.

Any number divisible by 2 (evens) are not prime. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Any number divisible by 2 (evens) are not prime.

So, 3 is the next prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 So, 3 is the next prime number.

Any number divisible by 3, is not a prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Any number divisible by 3, is not a prime number.

The next prime number is 5. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 The next prime number is 5.

Any number divisible by 5 is not a prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Any number divisible by 5 is not a prime number.

7 is the next prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 7 is the next prime number.

Any number divisible by 7 is not a prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Any number divisible by 7 is not a prime number.

11 is the next prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 11 is the next prime number.

Numbers divisible by 11 are already crossed out. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Numbers divisible by 11 are already crossed out.

13 is the next prime number. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 13 is the next prime number.

Numbers divisible by 13 are already crossed out. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Numbers divisible by 13 are already crossed out.

Next prime number is 17. All multiples crossed out. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Next prime number is 17. All multiples crossed out.

19 is the next prime number. All multiples crossed out. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 19 is the next prime number. All multiples crossed out.

The remaining numbers have no multiples uncrossed. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 The remaining numbers have no multiples uncrossed.

ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 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 8 2 5 2 4 2 2 24•5

24•5 24•5 PRIME FACTORIZATION 80 2 80 40 2 10 8 20 2 2 5 2 4 2 10 2 2 Write the prime factorization of 80. FACTOR TREE FACTOR TREE INVERTED DIVISION Start with smallest prime number 80 2 80 40 2 10 8 20 2 2 5 2 4 2 10 2 2 5 5 1 24•5 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? 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 + 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)

CMF 3x 5x x is a factor in common to both x is a monomial So, x is a FACTORING USING THE DISTRIBUTIVE PROPERTY If we take a look at two expressions: 3x 5x and x is a factor in common to both x is a monomial So, x is a Common Monomial Factor of 3x and 5x. Common Monomial Factor 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)

3m + 12 = 3(m + 4) 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)

m2 – 8m = m(m – 8) FACTORING PRACTICE Factor: m2 – 8m Step 1: What is the CMF? m Step 2: m times ? = m2 m(m Step 3: m times ? = -8m m(m – 8) m2 – 8m = m(m – 8)

10x2y + 5xy + 15y = 5y(2x2 – x + 3) FACTORING PRACTICE Factor: 10x2y – 5xy + 15y Step 1: What is the CMF? 5y Step 2: 5y times ? = 10x2y 5y(2x2 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

1 3a2 – 7b2 so, the expression is a prime polynomial. 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 –6cd2

For any real numbers a and b, if ab = 0, then either a = 0 or b= 0. 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.

STEP 1: Set equation equal to zero 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

STEP 4: Solve each equation SOLVING EQUATIONS Solve: 3m2 + 12m = 3m 3m2 + 12m = 3m STEP 1: Set equation = 0 -3m -3m 3m2 + 9m = 0 3m(m + 3) = 0 STEP 2: Factor left side 3m = 0 or m + 3 = 0 STEP 3: Set each factor = 0 3 3 -3 -3 m = 0 or m = -3 STEP 4: Solve each equation

STEP 4: Solve each equation SOLVING EQUATIONS Solve: 6x2 = -8x 6x2 = -8x STEP 1: Set equation = 0 +8x +8x 6x2 + 8x = 0 2x(3x + 4) = 0 STEP 2: Factor left side 2x = 0 or 3x + 4 = 0 STEP 3: Set each factor = 0 2 2 -4 -4 3x = -4 STEP 4: Solve each equation 3 3 x = 0 or x =