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Unformatted text preview: Math 135: Lecture 12: Modular Arithmetic Definition 12.1. The congruence class modulo m of the integer a is the set of integers [ a ] m = { x Z  x a (mod m ) } Example 12.2. When m = 4 [0] 4 = { x Z  x (mod 4) } = { ..., 8 , 4 , , 4 , 8 ,... } = { 4 k  k Z } [1] 4 = { x Z  x 1 (mod 4) } = { ..., 7 , 3 , 1 , 5 , 9 ,... } = { 4 k + 1  k Z } [2] 4 = { x Z  x 2 (mod 4) } = { ..., 6 , 2 , 2 , 6 , 10 ,... } = { 4 k + 2  k Z } [3] 4 = { x Z  x 3 (mod 4) } = { ..., 5 , 1 , 3 , 8 , 11 ,... } = { 4 k + 3  k Z } Note that congruence classes have more than one representation. In the example above, and, in fact [0] 4 has representations. Defining Z m We define Z m to be the set of m congruence classes Z m = { [0] , [1] , [2] ,..., [ m 1] } and we define two operations on Z m , addition and multiplication , as follows: [ a ] + [ b ] = [ a + b ] [ a ] [ b ] = [ a b ] Example 12.3 (Addition in Z 4 ) . + [0] [1] [2] [3] [0] [0] [1] [2] [3] [1] [1] [2] [3] [0] [2] [2] [3] [0] [1] [3] [3] [0] [1] [2] Example 12.4 (Multiplication in Z 4 ) . [0] [1] [2] [3] [0] [0] [0] [0] [0] [1] [0] [1] [2] [3] [2] [0] [2] [0] [2] [3] [0] [3] [2] [0] Exercise 12.5. Complete the addition table in Z 5 + [0] [1] [2] [3] [4] [0] [1] [2] [3] [4] Exercise 12.6. Complete the multiplication table in Z 5 + [0] [1] [2] [3] [4] [0] [1] [2] [3] [4] 1 Math 135: Lecture 12: Modular Arithmetic Zero in Z m From the definition of addition and multiplication in Z m [ a ] Z m , [ a ] Z m , One in Z m From our definition of multiplication in Z m...
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This note was uploaded on 10/27/2011 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.
 Fall '08
 ANDREWCHILDS
 Math, Congruence, Integers

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