Question

Write the \({ }_{7}\) multiplication table for \(Z_{7}\).

Short answer

The multiplication table for \( Z_7 \) is provided in the last step.

Step-by-step solution

Understanding the Set

First, identify the elements of the set \( Z_7 \), which consists of integers modulo 7. This set includes the elements \( \{0, 1, 2, 3, 4, 5, 6\} \).

Initialize the Multiplication Table

Set up a multiplication table with rows and columns labeled with the elements of \( Z_7 \) (i.e., \( 0, 1, 2, 3, 4, 5, 6 \)). This table will show the results of multiplying each pair of elements from \( Z_7 \).

Perform Modulo Multiplication

Calculate the product of each pair of numbers by multiplying them together and then taking the result modulo 7. For example, to calculate the product of 3 and 5, compute \( 3 \times 5 = 15 \) and then \( 15 \mod 7 = 1 \).

Fill Table Rows

Fill in the table for each row. For row of element \( a \):- \( 0 \times a \mod 7 = 0 \),- \( 1 \times a \mod 7 = a \),- Continue up to \( 6 \times a \mod 7 \). Calculate for all elements \( \{0, 1, 2, 3, 4, 5, 6\} \).

Write Down the Final Table

The final multiplication table should look as follows:\[\begin{array}{c|ccccccc}\times & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \2 & 0 & 2 & 4 & 6 & 1 & 3 & 5 \3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \4 & 0 & 4 & 1 & 5 & 2 & 6 & 3 \5 & 0 & 5 & 3 & 1 & 6 & 4 & 2 \6 & 0 & 6 & 5 & 4 & 3 & 2 & 1 \\end{array}\]

Conclusion

The multiplication table for \( Z_7 \) shows how each pair of numbers multiplies under modulo 7 conditions. Each entry in the table is calculated as \( (a \times b) \mod 7 \).

Key concepts

Multiplicative Group

A multiplicative group is a fundamental concept in algebra, particularly in the field of group theory. It involves a set of elements paired with a multiplication operation that satisfies certain properties. These properties include:

• Closure: The product of any two elements in the group is also an element in the group.
• Associativity: The way in which elements are grouped in multiplication does not affect the result.
• Identity element: There is an element, usually denoted as 1, such that multiplying any element by 1 leaves it unchanged.
• Inverses: For every element in the group, there must be another element that when multiplied with it results in the identity element.

The context of modulo arithmetic can introduce properties specific to groups like the multiplicative group of integers modulo a prime number, in this case, 7. Here, all non-zero elements possess a modular inverse under multiplication, making them suitable candidates to form a multiplicative group.

Modular Multiplication

Modular multiplication is a key operation in modular arithmetic, which simplifies calculations by "wrapping around" a certain number, known as the modulus. To perform modular multiplication:

• Multiply the numbers as you would in regular arithmetic.
• Apply the modulus by taking the remainder when the product is divided by the modulus.

For instance, when calculating the product of 3 and 5 under modulo 7, use the following steps: 1. Compute the product: 3 multiplied by 5 equals 15. 2. Apply the modulus: 15 mod 7 is 1, as 15 divided by 7 leaves a remainder of 1. This approach keeps numbers within a set range, simplifying computations significantly in many scenarios. In the exercise solution, this concept is illustrated through the multiplication table for the set of integers modulo 7, where each element represents the result of multiplying and then applying modulus.

Discrete Mathematics

Discrete mathematics is the branch of mathematics dealing with countable, distinct elements and encompasses topics such as integer arithmetic, graph theory, and combinatorics. It stands in contrast to continuous mathematics, which deals with continuously varying quantities. In the context of the multiplication table for integers modulo 7, discrete mathematics helps address concerns related to:

• Modular Arithmetic: Understanding calculations that involve whole numbers and their remainders after division.
• Structures: Working with sets of discrete elements, such as integers in modulo arithmetic, which is a typical discrete set.
• Number Theory: Explores properties such as congruences and relations among integers that underpin modular systems.

The study of such topics aids in solving practical problems in computer science, cryptography, and telecommunications, making discrete mathematics an invaluable tool in the broader field of mathematics and applied sciences.