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## ELEMENTARY NUMBER THEORY

by: Cassidy Grimes

29

0

2

# ELEMENTARY NUMBER THEORY MATH 580

Cassidy Grimes

GPA 3.51

M. Boylan

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COURSE
PROF.
M. Boylan
TYPE
Study Guide
PAGES
2
WORDS
KARMA
50 ?

## Popular in Mathematics (M)

This 2 page Study Guide was uploaded by Cassidy Grimes on Monday October 26, 2015. The Study Guide belongs to MATH 580 at University of South Carolina - Columbia taught by M. Boylan in Fall. Since its upload, it has received 29 views. For similar materials see /class/229516/math-580-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
Math 580 Exam 1 Information 91809 LC 121 125 215 Exam 1 will be based on 0 Sections 11 12 22 23 24 25 o The corresponding assigned homework problems see httpwwwmathscedub0ylanSCCOurses580Fa09580html At minimum you need to understand how to do the homework problems 0 Lecture notes 821 911 Topic List not necessarily comprehensive You will need to know theorems results and de nitions from class 11 Mathematical Induction How does a proof by induction proceed It consists of a base case and an induction step What is the well ordering principle 12 The Binomial Theorem Important formulas quot21k712 3 The Binomial Theorem Let n E N Then 96 y Zn llWk k0 1 Let0 k g n Then 2 Pascal s Identity We saw lots of ways to get identities by substituting speci c values for z and y 22 The Division Algorithm The division algorithm says If a b E Z with b 7 0 then there are unique integers q and r such that abqr 0 rltlbl Here q is for qu0tient and r is for remainder We considered applications in lecture and homework 23 The Greatest Common Divisor Key Facts 1 a divides b a l b if and only if there is a k E Z with ak b We discussed basic divisibility properties These include for example a lfa bthen lal b If a l b and a l c then for all integers z and y we have a l by by D How does one de ne the greatest common divisor god of a and b Key properties of the gcd a Let d gcda b Then there are integers z and y for which ax by d b Every linear combination ax by of a and b is a multiple of d gcda b c If gcda b 1 then a and b are coprime or relatively prime We have gcda b 1 if and only if there are integers z and y with ax by 1 This gives us a useful method for showing that gcda b 1 All we have to do is to exhibit integers z and y with ax by l or show that such z and y exist d If gcdab d then god 3 g 1 e lfa c b l c and gcdab 1 then ab c f lfa be and gcdab 1 then a l c 24 The Euclidean Algorithm The Euclidean algorithm is a step by step process used to compute d gcda b We did examples in class There are also examples in the homework Once we compute d gcda b we can nd z y E Z for which ax by d To do this we work backwards through the Euclidean algorithm at each step we solve for the remainder in the division algorithm and substitute A companion to the greatest common divisor of a and b is the least common multiple of a and b we denote it by lcma b How is it de ned The key formula is ab 1 b 7 0 gcdlta7bgt 25 The Diophantine Equation ax by 0 Our goal is to 1 Determine when this equation has solutions in integers z y 2 When it has integer solutions nd them all Key theorem Let d gcda b Then ax by c has a solution in integers if and only if d l 0 Moreover if d l c and zo yo is a solution in integers then all solutions in integers are given by b a x int iit VtEZ lt 0 d y 90 d Some problems ask you to nd all solutions z y with certain constraints For example if we want to nd all non negative solutions we solve for t in the following system of inequalities 7 b tgt0 7 atgt0 yix 7 7 77 0 d 7 71 90 d 7

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