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

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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

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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.

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

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