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Modular arithmetic Given any two integers a, b we define their addition mod n as (a+b) mod n, and denote this integer by a + b. That is (a+b)=(a+b) mod n Example: If a = 14, b = 5, and n = 6, then to compute 14 +6 5, we divide 14+5 = 19 by 6, which gives a quotient of 3 and a remainder of 1 since 19 = 3.6+1. Thus: 14 +65 = 1. Task 10 (5 Marks). Let n > 1 be any natural number and let Z = {0, 1, 2,...,n−1}. We will consider the mathematical structure (Zn,+n), which is called the integers modulo n. Show that: 1. Zn is closed under +n, 2. + has an identity element in Zn, 3. + is commutative, 4. + is associative, 5. every element in Zn has a +n-inverse.

Modular arithmetic Given any two integers a, b we define their addition mod n as (a+b) mod n, and denote this integer by a + b. That is (a+b)=(a+b) mod n Example: If a = 14, b = 5, and n = 6, then to compute 14 +6 5, we divide 14+5 = 19 by 6, which gives a quotient of 3 and a remainder of 1 since 19 = 3.6+1. Thus: 14 +65 = 1. Task 10 (5 Marks). Let n > 1 be any natural number and let Z = {0, 1, 2,...,n−1}. We will consider the mathematical structure (Zn,+n), which is called the integers modulo n. Show that: 1. Zn is closed under +n, 2. + has an identity element in Zn, 3. + is commutative, 4. + is associative, 5. every element in Zn has a +n-inverse.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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Modular arithmetic Given any two integers a, b we define their addition mod n as (a+b) mod n, and denote this integer by a + b. That is (a+b)=(a+b) mod n Example: If a = 14, b = 5, and n = 6, then to compute 14 +6 5, we divide 14+5 = 19 by 6, which gives a quotient of 3 and a remainder of 1 since 19 = 3.6+1. Thus: 14 +65 = 1. Task 10 (5 Marks). Let n > 1 be any natural number and let Z = {0, 1, 2,...,n−1}. We will consider the mathematical structure (Zn,+n), which is called the integers modulo n. Show that: 1. Zn is closed under +n, 2. + has an identity element in Zn, 3. + is commutative, 4. + is associative, 5. every element in Zn has a +n-inverse.
Transcribed Image Text:Modular arithmetic Given any two integers a, b we define their addition mod n as (a+b) mod n, and denote this integer by a + b. That is (a+b)=(a+b) mod n Example: If a = 14, b = 5, and n = 6, then to compute 14 +6 5, we divide 14+5 = 19 by 6, which gives a quotient of 3 and a remainder of 1 since 19 = 3.6+1. Thus: 14 +65 = 1. Task 10 (5 Marks). Let n > 1 be any natural number and let Z = {0, 1, 2,...,n−1}. We will consider the mathematical structure (Zn,+n), which is called the integers modulo n. Show that: 1. Zn is closed under +n, 2. + has an identity element in Zn, 3. + is commutative, 4. + is associative, 5. every element in Zn has a +n-inverse.
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