NUMBER THEORY Chapter 1: The Integers

The Well-Ordering Property.

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

Divisibility.

divisors

Linear Combination

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

Division Algorithm

More exercise

More examples

More example

More examples

Prime Numbers

Lemma (?)

How many Primes?

GREATEST COMMON DIVISOR

Greatest Common Divisor

Example

Relatively Prime

Example No common factor other than 1.

Linear Combination

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

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

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)

More examples

EUCLIDEAN ALGORITHM Number Theory

Example

Extended Euclidean Algorithm

FUNDAMENTAL THEOREM OF ARITHMETIC Integers

Greatest Common Divisor

LINEAR DIOPHANTINE EQUATION Integers