The problem involves constructing an operation table for $+_6$, which represents addition modulo 6, on the set $Z_6 = \{ 0, 1, 2, 3, 4, 5 \}$. To create the table, follow these steps:

Steps to Fill the Table:

1. Perform modular addition $(a + b) \mod 6$ for every pair $a, b \in Z_6$.
2. Compute the remainder when $a + b$ is divided by 6 for each cell in the table.

I'll fill out the operation table for you:

Explanation:

Each entry in the table is computed as: $(a + b) \mod 6$ For example:

• At row 2, column 4: $2 + 4 = 6$, and $6 \mod 6 = 0$.
• At row 5, column 3: $5 + 3 = 8$, and $8 \mod 6 = 2$.

Would you like further clarification on modular arithmetic?

Here are 5 questions to deepen your understanding:

1. What is modular arithmetic, and why is it useful in mathematics?
2. How would the operation table change if we used $+_5$ instead of $+_6$?
3. Can modular addition form a group under certain conditions? Explain why or why not.
4. How does modular arithmetic relate to real-world applications such as cryptography or clock arithmetic?
5. What properties does the set $Z_6$ with operation $+_6$ satisfy (e.g., closure, commutativity)?

Tip: Modular arithmetic can often simplify problems in number theory by focusing only on remainders!