Need help with b,c,d if possible ### Topic: Modular Arithmetic and Groups#### Section 1.15Write out the following tables for \( \mathbb{Z} /m\mathbb{Z} \) and \( (\mathbb{Z}/m\mathbb{Z})^* \), as we did in Figures 1.4 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 Terms- **Addition and Multiplication Tables**: In modular arithmetic, these tables show the results of adding or multiplying elements together under a given modulus.- **Unit Group**: The set of elements in \( \mathbb{Z}/m\mathbb{Z} \) that have a multiplicative inverse, denoted \( (\mathbb{Z}/m\mathbb{Z})^* \).By completing these exercises, you will deepen your understanding of basic arithmetic operations in modular arithmetic and the structure of unit groups. ### Understanding Modular Arithmetic: Addition and Multiplication Tables Modulo 5#### Addition Table Modulo 5The table on the left displays the addition table modulo 5. In modular arithmetic, numbers wrap around after reaching a certain value, which in this case is 5. The table is organized as follows:- The top row and the leftmost column enumerate the numbers from 0 to 4.- The cells within the table represent the sum of the corresponding row and column numbers, reduced modulo 5.For example:- \( 1 + 1 = 2 \)- \( 2 + 3 \mod 5 = 5 \mod 5 = 0 \)- \( 4 + 4 \mod 5 = 8 \mod 5 = 3 \)| | 0 | 1 | 2 | 3 | 4 ||---|---|---|---|---|---|| 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 |#### Multiplication Table Modulo 5The table on the right is the multiplication table modulo 5. Similar to the addition table, this table displays the product of the numbers in the corresponding row and column, reduced modulo 5.For example:- \( 1 \times 2 = 2 \)- \( 2 \times 3 \mod 5 = 6 \mod 5 = 1 \)- \( 4 \times 4 \mod 5 = 16 \mod 5 = 1 \)| | 0 | 1 | 2 | 3 | 4 ||---|---|---|---|---|---|| 0 | 0 | 0 | 0 | 0 | 0 || 1 | 0 | 1 | 2 | 3 | 4 || 2 | 0 | 2 | 4 | 1 | 3 || 3 | 0 |

(b) Make addition and multiplication tables for Z/6Z. (c) Make a multiplication table for the unit group (Z/9Z)*. (d) Make a multiplication table for the unit group (Z/16Z)*.

(b) Make addition and multiplication tables for Z/6Z. (c) Make a multiplication table for the unit group (Z/9Z). (d) Make a multiplication table for the unit group (Z/16Z).

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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Question

Need help with b,c,d if possible

### Topic: Modular Arithmetic and Groups

#### Section 1.15
Write out the following tables for \( \mathbb{Z} /m\mathbb{Z} \) and \( (\mathbb{Z}/m\mathbb{Z})^* \), as we did in Figures 1.4 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 Terms

- **Addition and Multiplication Tables**: In modular arithmetic, these tables show the results of adding or multiplying elements together under a given modulus.

- **Unit Group**: The set of elements in \( \mathbb{Z}/m\mathbb{Z} \) that have a multiplicative inverse, denoted \( (\mathbb{Z}/m\mathbb{Z})^* \).

By completing these exercises, you will deepen your understanding of basic arithmetic operations in modular arithmetic and the structure of unit groups.

### Understanding Modular Arithmetic: Addition and Multiplication Tables Modulo 5

#### Addition Table Modulo 5

The table on the left displays the addition table modulo 5. In modular arithmetic, numbers wrap around after reaching a certain value, which in this case is 5. The table is organized as follows:

- The top row and the leftmost column enumerate the numbers from 0 to 4.
- The cells within the table represent the sum of the corresponding row and column numbers, reduced modulo 5.

For example:
- \( 1 + 1 = 2 \)
- \( 2 + 3 \mod 5 = 5 \mod 5 = 0 \)
- \( 4 + 4 \mod 5 = 8 \mod 5 = 3 \)

| | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 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 |

#### Multiplication Table Modulo 5

The table on the right is the multiplication table modulo 5. Similar to the addition table, this table displays the product of the numbers in the corresponding row and column, reduced modulo 5.

For example:
- \( 1 \times 2 = 2 \)
- \( 2 \times 3 \mod 5 = 6 \mod 5 = 1 \)
- \( 4 \times 4 \mod 5 = 16 \mod 5 = 1 \)

| | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 |
| 2 | 0 | 2 | 4 | 1 | 3 |
| 3 | 0 |

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