36 Modular Arithmetic

Transcription

1 36 Modular Arithmetic Tom Lewis Fall Term 2010 Tom Lewis () 36 Modular Arithmetic Fall Term / 10

2 Outline 1 The set Z n 2 Addition and multiplication 3 Modular additive inverse 4 Modular multiplicative inverse 5 What are the invertible elements of Z n? Tom Lewis () 36 Modular Arithmetic Fall Term / 10

3 The set Z n Given an integer n 1, let Z n = {0, 1, 2, 3,, n 1}; We call this the set of integers modulo n. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

4 The set Z n Given an integer n 1, let Z n = {0, 1, 2, 3,, n 1}; We call this the set of integers modulo n. Note It is helpful to think of these as representatives of the equivalence classes modulo n of Z. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

5 Addition and multiplication Let a, b Z n. We define a b = (a + b) mod n a b = (a b) mod n Tom Lewis () 36 Modular Arithmetic Fall Term / 10

6 Addition and multiplication Let a, b Z n. We define a b = (a + b) mod n a b = (a b) mod n Problem Construct addition and multiplication tables for Z 3 and Z 4. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

7 Addition and multiplication Theorem Let n 2 be an integer. Let a, b, and c be elements of Z n. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

8 Addition and multiplication Theorem Let n 2 be an integer. Let a, b, and c be elements of Z n. Commutative a b = b a and a b = b a Tom Lewis () 36 Modular Arithmetic Fall Term / 10

9 Addition and multiplication Theorem Let n 2 be an integer. Let a, b, and c be elements of Z n. Commutative a b = b a and a b = b a Associative a (b c) = (a b) c and a (b c) = (a b) c Tom Lewis () 36 Modular Arithmetic Fall Term / 10

10 Addition and multiplication Theorem Let n 2 be an integer. Let a, b, and c be elements of Z n. Commutative a b = b a and a b = b a Associative a (b c) = (a b) c and a (b c) = (a b) c Identity a 0 = 0 and a 1 = a Tom Lewis () 36 Modular Arithmetic Fall Term / 10

11 Addition and multiplication Theorem Let n 2 be an integer. Let a, b, and c be elements of Z n. Commutative a b = b a and a b = b a Associative a (b c) = (a b) c and a (b c) = (a b) c Identity a 0 = 0 and a 1 = a Distributive a (b c) = (a b) (a c) Tom Lewis () 36 Modular Arithmetic Fall Term / 10

12 Modular additive inverse Theorem Given a Z n, there exists a unique x Z n such that a x = 0 mod n Tom Lewis () 36 Modular Arithmetic Fall Term / 10

13 Modular additive inverse Theorem Given a Z n, there exists a unique x Z n such that a x = 0 mod n Given a Z n, let a denote the unique element such that a is called the additive inverse of a. a ( a) = 0 mod n Tom Lewis () 36 Modular Arithmetic Fall Term / 10

14 Modular additive inverse Given a, b Z n, define a b = a ( b) Tom Lewis () 36 Modular Arithmetic Fall Term / 10

15 Modular additive inverse Given a, b Z n, define Problem a b = a ( b) Tom Lewis () 36 Modular Arithmetic Fall Term / 10

16 Modular additive inverse Given a, b Z n, define a b = a ( b) Problem Evaluate 8 and compute 3 8 in Z 11. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

17 Modular additive inverse Given a, b Z n, define a b = a ( b) Problem Evaluate 8 and compute 3 8 in Z 11. Evaluate 8 and compute 3 8 in Z 15. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

18 Modular multiplicative inverse Let a Z n. A reciprocal of a is a number b Z n such that a b = 1. A number that has a reciprocal is called invertible. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

19 Modular multiplicative inverse Let a Z n. A reciprocal of a is a number b Z n such that a b = 1. A number that has a reciprocal is called invertible. Problem Tom Lewis () 36 Modular Arithmetic Fall Term / 10

20 Modular multiplicative inverse Let a Z n. A reciprocal of a is a number b Z n such that a b = 1. A number that has a reciprocal is called invertible. Problem Find a reciprocal of 3 in Z 7. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

21 Modular multiplicative inverse Let a Z n. A reciprocal of a is a number b Z n such that a b = 1. A number that has a reciprocal is called invertible. Problem Find a reciprocal of 3 in Z 7. Find a reciprocal of 5 in Z 6. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

22 Modular multiplicative inverse Let a Z n. A reciprocal of a is a number b Z n such that a b = 1. A number that has a reciprocal is called invertible. Problem Find a reciprocal of 3 in Z 7. Find a reciprocal of 5 in Z 6. Show that 2 is not invertible in Z 6. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

23 Modular multiplicative inverse Theorem If a is invertible in Z n, then it has a unique inverse in Z n, denoted by a 1. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

24 Modular multiplicative inverse Theorem If a is invertible in Z n, then it has a unique inverse in Z n, denoted by a 1. (Division) Let n be a positive integer and let b be an invertible element of Z n. Let a Z n be arbitrary. Then a/b is defined by a b 1. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

25 Modular multiplicative inverse Theorem If a is invertible in Z n, then it has a unique inverse in Z n, denoted by a 1. (Division) Let n be a positive integer and let b be an invertible element of Z n. Let a Z n be arbitrary. Then a/b is defined by a b 1. Problem Compute 3/5 in Z 6. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

26 What are the invertible elements of Z n? Theorem Let n be a positive integer and let p Z n. p and n are relatively prime if and only if p is invertible in Z n. Tom Lewis () 36 Modular Arithmetic Fall Term / 10

27 What are the invertible elements of Z n? Theorem Let n be a positive integer and let p Z n. p and n are relatively prime if and only if p is invertible in Z n. Problem Evaluate 81/35 in Z 144. Tom Lewis () 36 Modular Arithmetic Fall Term / 10