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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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