modularkey - McCombs Math 381 Modular Arithmetic Basic Idea...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Modular Arithmetic 1 Basic Idea: Given an integer n ! 2 , we want to classify the set of integers according to the reminder obtained when dividing by n . Special Notation: ! a , b " Z , if n | a ! b ( ) , we write a ! b mod n ( ) , and say that “ a is congruent to b , mod n .” In other words, a ! b mod n ( ) means that a and b yield the same reminder when divided by n . In other words, a ! b mod n ( ) means that a = nk + b for some integer k . The Set of Integers mod n : Z n = 0,1,2,3,4,. .. n ! 1 { } Important Theorems: Theorem: Given , a b ! Z , with a and b not both 0. If ( ) gcd , d a b = , then there exist , x y ! Z such that d ax by = + . Theorem: Given , a b ! Z , with a and b not both 0. If ( ) gcd , d a b = , then is the smallest positive integer that can be expressed as the linear combination d ax by = + , where , x y ! Z . Theorem: Integers a and b are relatively prime if and only if there exist , x y ! Z such that 1 ax by + = . Modular Arithmetic Within the set Z n , we can perform standard arithmetic such as addition, subtraction, and multiplication. However, we must know how to “reduce mod n .” Given a , b ! Z n , 1. a ! b " a + b ( ) mod n ( ) 2. a ! b " a # b ( ) mod n ( ) Theorem: Given a ! Z n , a ! 1 (the multiplicative inverse of
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/16/2011 for the course MATH 381 taught by Professor Na during the Summer '11 term at University of North Carolina School of the Arts.

Page1 / 4

modularkey - McCombs Math 381 Modular Arithmetic Basic Idea...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online