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Math135Lecture12StudentNotes

# Math135Lecture12StudentNotes - Math 135 Lecture 12 Modular...

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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 0 (mod 4) } = { . . . , - 8 , - 4 , 0 , 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

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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 [ a ] Z m , Identity Given a set and an operation, an identity is, informally, “something that does nothing”. More formally, given a set S and an operation designated by , an identity is an element e S so that a S, a e = a Example 12.7. Set: integers; Operation: addition; Identity: Set: Q - { 0 }
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Math135Lecture12StudentNotes - Math 135 Lecture 12 Modular...

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