Presentation is loading. Please wait.

Presentation is loading. Please wait.

Quotient-Remainder Theory, Div and Mod 1. Exercise 2.

Similar presentations


Presentation on theme: "Quotient-Remainder Theory, Div and Mod 1. Exercise 2."— Presentation transcript:

1 Quotient-Remainder Theory, Div and Mod 1

2 Exercise 2

3 3

4 Floor & Ceiling 4 Definition:

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

6 Summary of Chapter 4 6

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

8 8

9 9

10 10

11 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.

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

13 Greatest Common Divisor (GCD) Definition 13

14 Greatest Common Divisor (GCD) 14

15 Greatest Common Divisor (GCD) 15

16 Greatest Common Divisor (GCD) 16

17 Greatest Common Divisor (GCD) 17

18 The Euclidean Algorithm 18

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

20 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

21 An alternative to Euclidean Algorithm 21

22 An alternative to Euclidean Algorithm 22

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

24 Homework #5 Problems 24

25 Exercise 25

26 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


Download ppt "Quotient-Remainder Theory, Div and Mod 1. Exercise 2."

Similar presentations


Ads by Google