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(a) Make addition and multiplication tables for Z/3Z.

(a) Make addition and multiplication tables for Z/3Z.

Advanced Engineering Mathematics
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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## Modular Arithmetic Tables ### Addition and Multiplication Tables Modulo 5 In modular arithmetic, numbers wrap around after reaching a certain value — the modulus. Here, we explore addition and multiplication modulo 5. These tables illustrate the results of these operations: #### Addition Table Modulo 5 This table shows the results of adding numbers from 0 to 4 with each other modulo 5. \[ \begin{array}{c|ccccc} + & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 \\ 1 & 1 & 2 & 3 & 4 & 0 \\ 2 & 2 & 3 & 4 & 0 & 1 \\ 3 & 3 & 4 & 0 & 1 & 2 \\ 4 & 4 & 0 & 1 & 2 & 3 \\ \end{array} \] #### Multiplication Table Modulo 5 This table shows the results of multiplying numbers from 0 to 4 with each other modulo 5. \[ \begin{array}{c|ccccc} \cdot & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 \\ 2 & 0 & 2 & 4 & 1 & 3 \\ 3 & 0 & 3 & 1 & 4 & 2 \\ 4 & 0 & 4 & 3 & 2 & 1 \\ \end{array} \] **Figure 1.4:** Addition and multiplication tables modulo 5 Each cell in these tables represents the result of the operation (either addition or multiplication) between the row and column headers, reduced modulo 5. For example, in the addition table, \(3 + 4 \equiv 2 \mod 5\), and in the multiplication table, \(2 \cdot 3 \equiv 1 \mod 5\). These tables are useful for understanding the structure and properties of modular arithmetic, which is widely used in fields such as cryptography, computer science, and number theory.
Transcribed Image Text:## Modular Arithmetic Tables ### Addition and Multiplication Tables Modulo 5 In modular arithmetic, numbers wrap around after reaching a certain value — the modulus. Here, we explore addition and multiplication modulo 5. These tables illustrate the results of these operations: #### Addition Table Modulo 5 This table shows the results of adding numbers from 0 to 4 with each other modulo 5. \[ \begin{array}{c|ccccc} + & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 \\ 1 & 1 & 2 & 3 & 4 & 0 \\ 2 & 2 & 3 & 4 & 0 & 1 \\ 3 & 3 & 4 & 0 & 1 & 2 \\ 4 & 4 & 0 & 1 & 2 & 3 \\ \end{array} \] #### Multiplication Table Modulo 5 This table shows the results of multiplying numbers from 0 to 4 with each other modulo 5. \[ \begin{array}{c|ccccc} \cdot & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 \\ 2 & 0 & 2 & 4 & 1 & 3 \\ 3 & 0 & 3 & 1 & 4 & 2 \\ 4 & 0 & 4 & 3 & 2 & 1 \\ \end{array} \] **Figure 1.4:** Addition and multiplication tables modulo 5 Each cell in these tables represents the result of the operation (either addition or multiplication) between the row and column headers, reduced modulo 5. For example, in the addition table, \(3 + 4 \equiv 2 \mod 5\), and in the multiplication table, \(2 \cdot 3 \equiv 1 \mod 5\). These tables are useful for understanding the structure and properties of modular arithmetic, which is widely used in fields such as cryptography, computer science, and number theory.
### Topic 1.15 #### Instruction: Write out the following tables for \( \mathbb{Z} / m \mathbb{Z} \) and \( (\mathbb{Z} / m \mathbb{Z})^* \), as we did in Figures 1.1 and 1.5. ### Exercises (a) Make addition and multiplication tables for \( \mathbb{Z} / 3\mathbb{Z} \). (b) Make addition and multiplication tables for \( \mathbb{Z} / 6\mathbb{Z} \). (c) Make a multiplication table for the unit group \( (\mathbb{Z} / 9\mathbb{Z})^* \). (d) Make a multiplication table for the unit group \( (\mathbb{Z} / 16\mathbb{Z})^* \). ### Explanation of the Task: 1. **Addition and Multiplication Tables for \( \mathbb{Z} / m \mathbb{Z} \)**: - **Addition Table**: This table shows the result of adding any two elements in the set \( \mathbb{Z} / m \mathbb{Z} \). - **Multiplication Table**: This table demonstrates the product of any two elements in the set \( \mathbb{Z} / m \mathbb{Z} \). 2. **Unit Group \( (\mathbb{Z} / m \mathbb{Z})^* \)**: - The unit group is the set of elements in \( \mathbb{Z} / m \mathbb{Z} \) that are relatively prime to \( m \). To complete these exercises, you should create addition and multiplication tables similar to what you have seen in Figures 1.1 and 1.5 in your textbook. This involves listing all possible sums and products within the specified sets and correctly formatting them into tables.
Transcribed Image Text:### Topic 1.15 #### Instruction: Write out the following tables for \( \mathbb{Z} / m \mathbb{Z} \) and \( (\mathbb{Z} / m \mathbb{Z})^* \), as we did in Figures 1.1 and 1.5. ### Exercises (a) Make addition and multiplication tables for \( \mathbb{Z} / 3\mathbb{Z} \). (b) Make addition and multiplication tables for \( \mathbb{Z} / 6\mathbb{Z} \). (c) Make a multiplication table for the unit group \( (\mathbb{Z} / 9\mathbb{Z})^* \). (d) Make a multiplication table for the unit group \( (\mathbb{Z} / 16\mathbb{Z})^* \). ### Explanation of the Task: 1. **Addition and Multiplication Tables for \( \mathbb{Z} / m \mathbb{Z} \)**: - **Addition Table**: This table shows the result of adding any two elements in the set \( \mathbb{Z} / m \mathbb{Z} \). - **Multiplication Table**: This table demonstrates the product of any two elements in the set \( \mathbb{Z} / m \mathbb{Z} \). 2. **Unit Group \( (\mathbb{Z} / m \mathbb{Z})^* \)**: - The unit group is the set of elements in \( \mathbb{Z} / m \mathbb{Z} \) that are relatively prime to \( m \). To complete these exercises, you should create addition and multiplication tables similar to what you have seen in Figures 1.1 and 1.5 in your textbook. This involves listing all possible sums and products within the specified sets and correctly formatting them into tables.
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