Section 4-1 Divisibility. Even integers can be represented by the algebraic expression: Odd integers can be represented by the algebraic expression:

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1 Section 4-1 Divisibility Even and Odd Numbers: A number is even if: It has 2 as a factor It is divisible by 2 (when divided by 2 the remainder is 0) A number is odd if: It does not have a factor of 2 When it is divided by 2 it has a remainder of 1. Even integers can be represented by the algebraic expression: Odd integers can be represented by the algebraic expression: Odd + Odd = Even + Even = Odd + Even = Definition of Divides : If a, b are whole numbers, then b divides a, denoted b a, if and only if there is a unique whole number q such that a = bq. (in other words, if a is a multiple of b). b will go into a with no remainder. If b a, then b is a factor of a. If b a, then b is a divisor of a. If b a, then a is a multiple of b If b a, then a is divisible by b If b does not divide a, then we write: Theorems: (For the following theorems, a, b, d, and n are whole numbers) 1. If d a, then d na (if d divides a, then d will divide any multiple of a) 2. If d a and d b then d (a ± b) (if d divides a and b, then d will divide their sum or difference) 3. If d a and d b then d (a ± b) (if d divides a but not b, then d will not divide their sum or difference) 1

2 Examples: True or False (for all of these examples, let a be any whole number) a) 5 15 b) 0 3 c) 8 4 d) 2 4a e) If 3 a, then 3 5a. f) If b a and b c, then b ac. g) If 7 a, and 7 b, then 7 (a + b) h) If 7 (a + b), then 7 a, and 7 b. 2

3 i) If 2 a, and 3 a, then 6 a. j) If 9 a, then 3 a DIVISIBILITY RULES A whole number is divisible by 2 if and only if the ones digit is even. Ex: Is 632 divisible by 2? Is 1,157 divisible by 2? A whole number is divisible by 5 if and only if the ones digit is 0 or 5. Ex: Is 751 divisible by 5? Is 20,255 divisible by 5? A whole number is divisible by 10 if and only if the ones digit is 0. Ex: Is 11,760 divisible by 10? Is 213 divisible by 10? 3

4 A whole number is divisible by 4 if and only if the two rightmost digits of the number represent a number that is divisible by 4. Ex: Is 832 divisible by 4? Is 5,514 divisible by 4? A whole number is divisible by 8 if and only if the three rightmost digits of the number represent a number that is divisible by 8. Ex: Is 880 divisible by 8? Is 1,024 divisible by 8? A whole number is divisible by 3 if and only if the sum of its digits is divisible by 3. Ex: Is 739 divisible by 3? Is 28,641 divisible by 3? A whole number is divisible by 9 if and only if the sum of its digits is divisible by 9. Ex: Is 28,641 divisible by 9? Is 585 divisible by 9? 4

5 A whole number is divisible by 6 if and only if it is divisible by both 2 and 3. Ex: Is 486 divisible by 6? Is 952 divisible by 6? Example: Test the number 2,640 for divisibility by: 2 Yes/No 3 Yes/No 4 Yes/No 5 Yes/No 6 Yes/No 8 Yes/No 9 Yes/No 10 Yes/No 11 Yes/No Example: Fill in the blanks so that the number is divisible by 6 and 5: 6 3,41 5

6 Section 4-2 Prime and Composite Numbers Definitions: A natural number is a prime number if and only if it has. A natural number is a composite number if and only if it has. Examples: Prime or Composite? a) 3 b) 4 c) 8 d) 1 e) 2 Prime Factorization When you represent a number as the product of prime numbers. Using a Factor Tree: Using Division by Primes:

7 The Fundamental Theorem of Arithmetic Each composite number can be written as a unique product of prime factors. How Many Whole Number Divisors Will a Number Have? Consider the prime factorization of 24: 24 = The divisors of 2 3 are: 2 0, 2 1, 2 2, 2 3 (there are 4 divisors) The divisors of 3 1 are: 3 0, 3 1 (there are 2 divisors) Using the Fundamental Counting Principle, there will be a total of 4 2 (or 8) divisors. They are: 2 0 (1) 3 1 (3) 2 1 *3 1 (6) 2 1 (2) 2 2 *3 1 (12) 2 2 (4) 2 3 *3 1 (24) 2 3 (8) Example: How many divisors does the number 180 have? Example: How many divisors does the number 1,000,000 have? 7

8 The Sieve of Eratosthenes

9 Determining if a Number is Prime or Composite: Theorem: If n is a whole number greater than 1 and not divisible by any prime p such that p 2 n, then n is prime. To determine whether a number n is a prime number, list all the prime numbers p that satisfy the equation p 2 n. If none of those prime numbers divide n, then n is a prime number. Example: Is the number 103 prime? (hint: ) Example: Is the number 507 prime? (hint: ) 9

10 Section 4-3 Greatest Common Divisor and Least Common Multiple Greatest Common Divisor: The GCD of two whole numbers a and b is the greatest whole number that divides both a and b. Note: Two numbers that have no factors in common (other than 1) are called relatively prime Methods for Finding GCD: Intersection of Sets Method: List the divisors of each number, then look for the intersection of the two sets. The GCD is the largest divisor in the intersection set. Example: Find GCD(20, 42) = Divisors of 20 = Divisors of 42 = The largest divisor both sets have in common is. Example: Find GCD(40, 185) 10

11 Prime Factorization Method: Find the prime factorization of each number. Identify the prime factors that are common to both numbers. The product of these common prime factors is the GCD. Example: Find GCD(45, 75) = 45 = 75 = Example: Find GCD(735, 1575) Division-by-Primes Method: Divide each number by the prime factors that are common to both. The GCD is the product of all of the common factors. Example: Find GCD(108, 120) = Example: GCD(90, 105, 315) 11

12 Euclidean Algorithm If a and b are any whole numbers where b is not equal to 0 and a is greater than or equal to b, then GCD(a,b) = GCD(r,b), where r is the remainder when a is divided by b. Repeated use of this rule leads to the Euclidean Algorithm. Example: Use the Euclidean algorithm to find GCD (75, 45) Example: GCD (2924, 220) Least Common Multiple: The LCM of two whole numbers a and b is the smallest non-zero whole number that is simultaneously a multiple of a and a multiple of b. Methods for Finding LCM: Intersection of Set Method: List the multiples of each number, then look for the intersection of the two sets. The LCM is the smallest multiple in the intersection set. Example: Find LCM(12, 60) = Multiples of 12 = Multiples of 60 = The first multiple they both have in common is. Example: Find LCM(15, 40) 12

13 Prime Factorization Method: Find the prime factorization of each number. Take each prime factor that occurs in either number. Use the largest exponent on each factor. The product of these prime factors is the LCM. Example: LCM(18, 20) = 18 = 20 = Example: Find LCM(45, 75) Division-by-Primes Method: Divide each number by the prime factors that are common to either number. If a factor does not divide into a number, carry that number up. Keep dividing until you get 1 at the top. The LCM is the product of all of the prime factors. Example: Find LCM(108, 120) = Example: Find LCM(90, 105, 315) = 13

14 The GCD-LCM Product Method: a b = GCD(a, b) LCM(a, b) Find the GCD first (it s usually easier), then plug into the formula to find the LCM. Example: a b GCD(a, b) LCM(a, b) Example: Find LCM(50, 72) = The Venn Diagrams Method: Example: Consider the prime factorization of 12 and 30: Factors of 12 Factors of 30 Notice, the LCM is the of the two sets, and the GCD is the of the two sets. 14

15 Example: Find GCD (24, 60) and LCM (24, 60) Examples: a) What is the GCD of two even numbers? b) What is the GCD of two prime numbers? c) What is the LCM of two prime numbers? d) What is the GCD of two relatively prime numbers? e) What is the LCM of two relatively prime numbers? 15

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