1. CHAPTER 4
Number Theory

2. 4.1 Prime & Composite Numbers
Product = is the result of multiplying. For example, 6 is the product of 2 and 3
Multiple = is the product of itself and a natural number (a positive whole number greater than 0). For example, the multiples of 9 are 9, 18, 27, 36, 45, etc.
Factor = is any integer that divides another integer with no remainder. For example, 3 and 9 are factors of 27 (because 3 goes into 27 nine times)

3. Example 1 1 × (6) = 6 2 × (6) = 12 3 × (6) = 18 4 × (6) = 24
List the first four multiples of 6.
1 × (6) = 6
2 × (6) = 12
3 × (6) = 18
4 × (6) = 24

4. 4.1 Prime Factorization Divisibility Test (table also on page 137)
2 = the integer ends in an even digit 0, 2, 4, 6, or 8
3 = the sum of the integer’s digits is divisible by 3
4 = the number formed by the last 2 digits is divisible by 4
5 = the integer ends in 0 or 5
6 = the integer is divisible by both 2 and 3
8 = the number formed by the last 3 digits is divisible by 8
9 = the sum of the integer’s digits is divisible by 9
10 = the integer ends in 0

5. Example 2
Is 3 a factor of 117? Show answer
Yes, 3 is a factor of 117 because:
the sum of the integer’s digits is divisible by 3
3 117 39 9 27
1+1+7 = 9/3 = (yes 3)

6. Example 3
List in order all the factors of 12.
1, 2, 3, 4, 6, 12

7. 4.1 Prime & Composite Numbers
Prime Number = is a natural number only divisible by 1 and itself (examples: 2, 3, 5, 7)
Composite Number = is any natural number divisible by factors other than 1 and itself (examples: 4, 6, 8, 9, 10)
Neither = 0, 1

8. Topic: Prime and Composite Numbers
HOMEWORK
Topic: Prime and Composite Numbers
examples on pages
Assignment: Lesson 4.1
in book on pages
1-39 odd (20 total)

9. Topic: Prime and Composite Numbers
HOMEWORK
Topic: Prime and Composite Numbers
examples on pages
Assignment: Lesson 4.1
in book on pages
2-40 even (20 total)

11. 4.2 Prime Factorization
Prime Number = is a natural number only divisible by 1 and itself (examples: 2, 3, 5, 7)
Factor Tree Rules
always start with the smallest prime factors
only circle prime numbers
keep factoring numbers until all remaining numbers are prime numbers

12. Example 1 Write the prime factorization of 42. 42 = 2 • 3 • 7 2 • 21
2 • 21
3 • 7

13. Example 2 Find the number whose prime factorization is given. 23 • 3
8 • 3
= 24

14. Example 3 Solve the equation to find the missing prime factor.
x = 2

15. Topic: Prime Factorization
HOMEWORK
Topic: Prime Factorization
examples on pages
Assignment: Lesson 4.2
in book on page 143
1-35 odd (18 total)

16. Topic: Prime Factorization
HOMEWORK
Topic: Prime Factorization
examples on pages
Assignment: Lesson 4.2
in book on page 143
2-36 even (18 total)

17. 4.1 Prime Factorization Divisibility Test (table also on page 137)
2 = the integer ends in an even digit 0, 2, 4, 6, or 8
3 = the sum of the integer’s digits is divisible by 3
4 = the number formed by the last 2 digits is divisible by 4
5 = the integer ends in 0 or 5
6 = the integer is divisible by both 2 and 3
8 = the number formed by the last 3 digits is divisible by 8
9 = the sum of the integer’s digits is divisible by 9
10 = the integer ends in 0

19. 4.3 Greatest Common Factor (GCF)
Factor = is any integer that divides the given integer with no remainder
GCF = is the largest number that is common between 2 or more numbers
Relatively Prime = is when the only common number is 1 (ex. 13 and 17)

20. Example 1 What is the GCF of 12 and 60? 12 60 2 • 6 2 • 30 2 • 3
2 • 6
2 • 30
2 • 3
2 • 15
3 • 5
GCF = 2 • 2 • 3 = 12

21. Helpful Shortcuts
Check if the smaller of the two numbers is the GCF
Example 12 and 60
2) Check if the difference between the two numbers is the GCF
Example 75 and 90

22. Example 2 15x3y = 3 • 5 • x • x • x • y 21x2 = 3 • 7 • x • x
Find the GCF of 15x3y and 21x2.
15x3y = 3 • 5 • x • x • x • y
21x2 = 3 • 7 • x • x
GCF = 3 • x • x
= 3x2

23. Example 3 Find the GCF of the numbers given in factored form.
23 • 32 and 22 • 32
22 • 32
4 • 9
= 36

24. Topic: Greatest Common Factor
HOMEWORK
Topic: Greatest Common Factor
examples on pages
Assignment: Lesson 4.3
in book on page 147
3-33 odd (16 total)

25. Topic: Greatest Common Factor
HOMEWORK
Topic: Greatest Common Factor
examples on pages
Assignment: Lesson 4.3
in book on page 147
4-34 even (16 total)

26. Finding the GCF
Write the prime factorization (factor tree) for each number.
Identify the common prime factors for both numbers.
Find the product of the common prime factors.

28. 4.4 Least Common Multiple (LCM)
Least Common Multiple = is the smallest number that is a multiple of each number.

29. Example 1 List the multiples of 6 and 10 to find the LCM (book 1-6)
6: 6, 12, 18, 24, 30, 36, 42…
10: 10, 20, 30, 40, 50, 60, 70…
LCM = 30

30. Example 2 2 • 3 • 5 and 32 • 5 2 • 32 • 5 2 • 9 • 5 18 • 5 LCM = 90
To find the LCM use the highest power of each prime factor (book 7-16)
2 • 3 • 5 and 32 • 5
2 • 32 • 5
2 • 9 • 5
18 • 5
LCM = 90

31. Example 3
To find the LCM, use your factor tree to find the highest power of each prime factor (book 17-28) 18 24
LCM = 23 • 32
= 8 • 9
= 72
2 • 9
3 • 3
2 • 12
2 • 6
2 • 3
18 = 2 • 32
24 = 23 • 3

32. Example 4 9c4d5 12c3d7 LCM = 22 • 32 = 4 • 9 = 36c4d7 3 • 3 2 • 6
To find the LCM, use the highest power of each prime factor & variable (book 29-34)
9c4d5
12c3d7
LCM = 22 • 32
= 4 • 9
= 36c4d7
3 • 3
2 • 6
2 • 3
9 = 32
12 = 22 • 3

33. Topic: Least Common Multiple (LCM)
HOMEWORK
Topic: Least Common Multiple (LCM)
examples on pages
Assignment: Lesson 4.4
in book on pages
3-31 odd (15 total)

34. Topic: Least Common Multiple (LCM)
HOMEWORK
Topic: Least Common Multiple (LCM)
examples on pages
Assignment: Lesson 4.4
in book on pages
4-32 even (15 total)

35. Finding the LCM
Write the prime factorization (factor tree) for each number.
Identify ALL prime factors for both numbers.
Find the product using the highest power of each prime factor.

37. Distributive Property (day 1)
4.9 Factoring Using
Distributive Property (day 1)
Monomial = an algebraic expression consisting of one term
Binomial = an algebraic expression consisting of two terms
Trinomial = an algebraic expression consisting of three terms
Polynomial = an algebraic expression consisting of one or more monomials (each monomial is a term in a polynomial)

38. A polynomial CANNOT have a variable in the denominator
Monomial Binomial Trinomial 3x
3x x
3x x
A polynomial CANNOT have a variable in the denominator

39. Determine whether each is a polynomial and the type10 h² j
Yes, this is a monomial (constant)
No, this has a variable in the denominator
Yes, this is a monomial (a product of variables)
Yes, this is a monomial (single variable)

40. Determine whether each is a polynomial and the type
8x – 2x + 5
– y – 3
23abcd²
𝟑𝒙 𝒙
Yes, this is a trinomial (three terms)
Yes, this is a binomial (two terms)
Yes, this is a product of variables and nonnegative integer exponents (1 term)
Yes, this is a product of a real number multiplied by variables with nonnegative integer exponents (1 term)
No, one of the terms has a variable in the denominator

41. Steps to Solve 27y² + 18y 27y2 18y 3 • 9 • y • y 3 • 3 2 • 9 • y 3 • 3
Find the Greatest Common Factor (GCF)
Use the GCF to rewrite each term
27y² + 18y
27y2
18y
3 • 9 • y • y
3 • 3
2 • 9 • y
3 • 3
GCF = 3 • 3 • y = 9y
9y(3y + 2)

42. Topic: Factoring using Distributive Property
HOMEWORK
Topic: Factoring using Distributive Property
Assignment: Lesson 4.9
on worksheet
(17 total)

44. 4.9 Factoring a Polynomial by Grouping (day 2)
Must have at least 4 terms
Terms must have common factors that can be grouped together

45. 4.9 Factoring a Polynomial by Grouping
= 4qr + 8r + 3q + 6
= (4qr + 8r) + (3q + 6)
= 4r(q + 2) + 3(q + 2)
= (4r + 3)(q + 2)

46. 4.9 Factoring a Polynomial by Grouping
= rn + 5n – r – 5
= (rn + 5n) + (– r – 5)
= n(r + 5) – 1(r + 5)
= (n – 1)(r + 5)

47. Topic: Factoring using Distributive Property
HOMEWORK
Topic: Factoring using Distributive Property
Assignment: Lesson 4.9
on worksheet
(22 total)