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NUMBER THEORY Chapter 1: The Integers

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The Well-Ordering Property.

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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,…}

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Divisibility.

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divisors

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Linear Combination

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

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Division Algorithm

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More exercise

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More examples

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More example

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More examples

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Prime Numbers

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Lemma (?)

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How many Primes?

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GREATEST COMMON DIVISOR

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Greatest Common Divisor

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Example

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Relatively Prime

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Example No common factor other than 1.

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Linear Combination

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

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

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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)

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More examples

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EUCLIDEAN ALGORITHM Number Theory

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Example

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Extended Euclidean Algorithm

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FUNDAMENTAL THEOREM OF ARITHMETIC Integers

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Greatest Common Divisor

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LINEAR DIOPHANTINE EQUATION Integers

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