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Quotient-Remainder Theory, Div and Mod 1

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Exercise 2

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Floor & Ceiling 4 Definition:

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Relations between Proof by Contradiction and Proof by Contraposition 5 Sequence of steps Same sequence of steps

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Summary of Chapter 4 6

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How to proof: Direct proof? Proof by contradiction? Proof by contraposition? 7

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Property of a Prime Divisor Proposition 4.7.3 If a prime number divides an integer, then it does not divide the next successive integer.

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Infinitude of the Primes 12 Theorem 4.7.4 Infinitude of the Primes The set of prime numbers is infinite.

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Greatest Common Divisor (GCD) Definition 13

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Greatest Common Divisor (GCD) 14

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Greatest Common Divisor (GCD) 15

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Greatest Common Divisor (GCD) 16

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Greatest Common Divisor (GCD) 17

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The Euclidean Algorithm 18

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The Euclidean Algorithm - Exercise Use the Euclidean algorithm to find gcd(330, 156). 19

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The Euclidean Algorithm - Exercise Use the Euclidean algorithm to find gcd(330, 156). Solution: gcd(330,156) = gcd(156, 18) 330 mod 156 = 18 = gcd(18, 12) 156 mod 18 = 12 = gcd(12, 6) 18 mod 12 = 6 = gcd(6, 0) 12 mod 6 = 0 = 6 20

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An alternative to Euclidean Algorithm 21

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An alternative to Euclidean Algorithm 22

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Homework #5 Problems Converting decimal to rational numbers. 4.2.2: 4.6037 4.2.7: 52.4672167216721… 23

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Homework #5 Problems 24

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Exercise 25

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Homework #5 Problems Theorem: The sum of any two even integers equals 4k for some integer k. “Proof: Suppose m and n are any two even integers. By definition of even, m = 2k for some integer k and n = 2k for some integer k. By substitution, m + n = 2k + 2k = 4k. This is what was to be shown.” What’s the mistakes in this proof? 26

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