To solve the problems in $\mathbb{Z}/8$ (the integers modulo $8$), we perform arithmetic operations and reduce each result to its least positive representative modulo $8$.

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(a) $[7] + [6]$

1. Compute the sum: $7 + 6 = 13$
2. Reduce modulo $8$: $13 \mod 8 = 5$ Answer: $[7] + [6] = [5]$.

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(b) $[2] \cdot ([3] + [2])$

1. Compute the sum inside the parentheses: [ [3] + [2] = 3 + 2 = 5 ] $5 \mod 8 = 5 \quad (\text{already less than } 8)$ So, $[3] + [2] = [5]$.
2. Multiply by $[2]$: [ [2] \cdot [5] = 2 \cdot 5 = 10 ]
3. Reduce modulo $8$: $10 \mod 8 = 2$ Answer: $[2] \cdot ([3] + [2]) = [2]$.

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(c) $([4] + [2]) \cdot ([5] + [3])$

1. Compute the sum of the first pair: [ [4] + [2] = 4 + 2 = 6 ] $6 \mod 8 = 6 \quad (\text{already less than } 8)$ So, $[4] + [2] = [6]$.
2. Compute the sum of the second pair: [ [5] + [3] = 5 + 3 = 8 ] $8 \mod 8 = 0$ So, $[5] + [3] = [0]$.
3. Multiply the results: [ [6] \cdot [0] = 6 \cdot 0 = 0 ]
4. Reduce modulo $8$: $0 \mod 8 = 0$ Answer: $([4] + [2]) \cdot ([5] + [3]) = [0]$.

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Final Answers

(a) $[7] + [6] = [5]$
(b) $[2] \cdot ([3] + [2]) = [2]$
(c) $([4] + [2]) \cdot ([5] + [3]) = [0]$

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Would you like further clarification or details about modular arithmetic?

Here are 5 related questions for practice:

1. Compute $[5] \cdot [7]$ in $\mathbb{Z}/8$.
2. Find the additive inverse of $[3]$ in $\mathbb{Z}/8$.
3. Solve $[4] \cdot ([6] - [5])$ in $\mathbb{Z}/8$.
4. Determine $[2]^3 \mod 8$.
5. Verify if $[6]$ has a multiplicative inverse in $\mathbb{Z}/8$.

Tip: Remember, in modular arithmetic, addition, subtraction, and multiplication work the same way as usual arithmetic, but results are always reduced modulo $n$.