CHAPTER 4 Number Theory 4 1 Prime Composite

CHAPTER 4 Number Theory

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)

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

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

Example 2 Is 3 a factor of 117? Show answer 39 3 117 ü the sum of the 9 integer’s digits is 27 divisible by 3 27 1+1+7 = 9/3 = (yes 3) 0 Yes, 3 is a factor of 117 because:

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

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

HOMEWORK Topic: Prime and Composite Numbers examples on pages 136 -138 Assignment: Lesson 4. 1 in book on pages 139 -140 1 -39 odd (20 total)

HOMEWORK Topic: Prime and Composite Numbers examples on pages 136 -138 Assignment: Lesson 4. 1 in book on pages 139 -140 2 -40 even (20 total)

4. 2 Prime Factorization Prime Number = is a natural number only divisible by 1 and itself (examples: 2, 3, 5, 7) Factor Tree Rules 1) always start with the smallest prime factors 2) only circle prime numbers 3) keep factoring numbers until all remaining numbers are prime numbers

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

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

Example 3 Solve the equation to find the missing prime factor. 30 = x • 3 • 5 30 = x • 15 30 = 15 x 15 15 x=2

HOMEWORK Topic: Prime Factorization examples on pages 141 -143 Assignment: Lesson 4. 2 in book on page 143 1 -35 odd (18 total)

HOMEWORK Topic: Prime Factorization examples on pages 141 -143 Assignment: Lesson 4. 2 in book on page 143 2 -36 even (18 total)

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

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)

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

Helpful Shortcuts 1) 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

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

Example 3 Find the GCF of the numbers given in factored form. 3 2 2 • 3 and 2 • 3 2 2 2 • 3 4 • 9 = 36

HOMEWORK Topic: Greatest Common Factor examples on pages 144 -146 Assignment: Lesson 4. 3 in book on page 147 3 -33 odd (16 total)

HOMEWORK Topic: Greatest Common Factor examples on pages 144 -146 Assignment: Lesson 4. 3 in book on page 147 4 -34 even (16 total)

Finding the GCF 1. Write the prime factorization (factor tree) for each number. 2. Identify the common prime factors for both numbers. 3. Find the product of the common prime factors.

4. 4 Least Common Multiple (LCM) Least Common Multiple = is the smallest number that is a multiple of each number.

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

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

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

Example 4 To find the LCM, use the highest power of each prime factor & variable (book 29 -34) 4 5 9 c d 3 • 3 3 7 12 c d 2 • 6 2 • 3 9=3 2 2 12 = 2 • 3 2 2 LCM = 2 • 3 =4 • 9 4 7 = 36 c d

HOMEWORK Topic: Least Common Multiple (LCM) examples on pages 148 -151 Assignment: Lesson 4. 4 in book on pages 150 -151 3 -31 odd (15 total)

HOMEWORK Topic: Least Common Multiple (LCM) examples on pages 148 -151 Assignment: Lesson 4. 4 in book on pages 150 -151 4 -32 even (15 total)

Finding the LCM 1. Write the prime factorization (factor tree) for each number. 2. Identify ALL prime factors for both numbers. 3. Find the product using the highest power of each prime factor.

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)

Polynomial Monomial Binomial Trinomial 3 x 3 x + 5 x + 7 A polynomial CANNOT have a variable in the denominator

Determine whether each is a polynomial and the type • 10 Yes, this is a monomial (constant) • No, this has a variable in the denominator • h² Yes, this is a monomial (a product of variables) • j Yes, this is a monomial (single variable)

Determine whether each is a polynomial and the type • 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

Steps to Solve 1) Find the Greatest Common Factor (GCF) 2) Use the GCF to rewrite each term 27 y 2 27 y² + 18 y 3 • 9 • y 3 • 3 2 • 9 • y 3 • 3 GCF = 3 • y = 9 y 9 y(3 y + 2)