### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# ELEMENTARY NUMBER THEORY MATH 580

GPA 3.51

### View Full Document

## 29

## 0

## Popular in Course

## 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.

## Reviews for ELEMENTARY NUMBER THEORY

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.