Download presentation

Presentation is loading. Please wait.

Published byZoe Baker Modified about 1 year ago

1
NUMBER THEORY Chapter 1: The Integers

2
The Well-Ordering Property.

3
example Finite set – {1,2,3,4,5} – {2,4,6,7,15} – {101, 10001, 100001, 11, 111} Infinite set – {1,3,5,7,9,11,…} – {1,1,2,3,5,8,13,21,34,…}

4
Divisibility.

6
divisors

11
Linear Combination

14
Exercise If 7| 21 and 7|49, suggest 3 more integers divisible by 7.

15
Division Algorithm

17
More exercise

18
More examples

19
More example

20
More examples

22
Prime Numbers

25
Lemma (?)

27
How many Primes?

33
GREATEST COMMON DIVISOR

34
Greatest Common Divisor

35
Example

36
Relatively Prime

38
Example No common factor other than 1.

42
Linear Combination

44
Bezout’s theorem If a and b are integers, then there are integers m and n such that ma+nb=(a,b).

45
Corollary a and b are relatively prime if and only if there is integers a and b, ma+nb=1.

46
Interesting result a and b are relatively prime if and only if there is integers a and b, ma+nb=1. (na, nb)=n (a,b)

47
More examples

54
EUCLIDEAN ALGORITHM Number Theory

57
Example

58
Extended Euclidean Algorithm

59
FUNDAMENTAL THEOREM OF ARITHMETIC Integers

64
Greatest Common Divisor

71
LINEAR DIOPHANTINE EQUATION Integers

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google