1. 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.

2. PRIME NUMBERS ARE GREATER THAN 1.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
PRIME NUMBERS ARE GREATER THAN 1.

3. So, 2 is the smallest prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
So, 2 is the smallest prime number.

4. Any number divisible by 2 (evens) are not prime.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Any number divisible by 2 (evens) are not prime.

5. So, 3 is the next prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
So, 3 is the next prime number.

6. Any number divisible by 3, is not a prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Any number divisible by 3, is not a prime number.

7. The next prime number is 5.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
The next prime number is 5.

8. Any number divisible by 5 is not a prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Any number divisible by 5 is not a prime number.

9. 7 is the next prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
7 is the next prime number.

10. Any number divisible by 7 is not a prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Any number divisible by 7 is not a prime number.

11. 11 is the next prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
11 is the next prime number.

12. Numbers divisible by 11 are already crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Numbers divisible by 11 are already crossed out.

13. 13 is the next prime number.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
13 is the next prime number.

14. Numbers divisible by 13 are already crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Numbers divisible by 13 are already crossed out.

15. Next prime number is 17. All multiples crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
Next prime number is 17. All multiples crossed out.

16. 19 is the next prime number. All multiples crossed out.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
19 is the next prime number. All multiples crossed out.

17. The remaining numbers have no multiples uncrossed.
ERATOSTHENE’S SIEVE
FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
The remaining numbers have no multiples uncrossed.

18. ERATOSTHENE’S SIEVE FIND ALL THE PRIME NUMBERS BETWEEN 1 AND 100
The numbers circled are prime numbers.
Except for 1, all the rest are composite numbers.

19. PRIME FACTORIZATION
Write the prime factorization of 80.
FACTOR TREE 80 10 8 2 5 2 4 2 2
24•5

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. FLASH CARDS
WHAT IS THE GCF?
20 and 30 10

27. FLASH CARDS
WHAT IS THE GCF?
4x and 6y 2

28. FLASH CARDS
WHAT IS THE GCF?
6m and 12m 6m

29. FLASH CARDS
WHAT IS THE GCF?
8xy and 12xz 4x

30. FLASH CARDS
WHAT IS THE GCF?
10a2b and 14ab2
2ab

31. 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)

32. 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)

33. 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

34. 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)

35. 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)

36. 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)

37. 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)

38. 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)

39. FLASH CARDS
WHAT IS THE CMF?
6x + 15 3

40. FLASH CARDS
WHAT IS THE CMF?
12m2 – 8m 4m

41. 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.

42. FLASH CARDS
WHAT IS THE CMF?
– 4b3 + 8b2c – 6bc2 2b

43. FLASH CARDS
WHAT IS THE CMF?
a3b+ a2b2 – ab3 ab

44. FLASH CARDS
FILL IN THE BLANK?
ab(___) = 3ab2 3b

45. FLASH CARDS
FILL IN THE BLANK?
3m(___) = 6m2 2m

46. FLASH CARDS
FILL IN THE BLANK?
5xy(___) = 15x2y 3x

47. FLASH CARDS
FILL IN THE BLANK?
2cd(___) = –12c2d3
–6cd2

48. 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.

49. 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

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

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