Given the problem:

Let $a, b, c \in \mathbb{Z}$ (i.e., $a, b, c$ are integers), and it's stated that $a \mid b$ (a divides b) and $a \mid c$ (a divides c). We need to determine which of the following options is correct:

a) $a \mid (bc)$

Since $a \mid b$ and $a \mid c$, this implies that $b = a \cdot k_1$ and $c = a \cdot k_2$ for some integers $k_1, k_2$. Therefore, the product $bc = (a \cdot k_1)(a \cdot k_2) = a^2 \cdot (k_1 \cdot k_2)$, which clearly shows that $a$ divides $bc$ (since $a^2 \cdot (k_1 \cdot k_2)$ contains at least one factor of $a$).

Thus, option a) is correct.

b) $a \mid (b + c)$

Since $a \mid b$ and $a \mid c$, we know that $b = a \cdot k_1$ and $c = a \cdot k_2$. Thus, $b + c = a \cdot k_1 + a \cdot k_2 = a \cdot (k_1 + k_2)$, which shows that $a$ divides $b + c$.

Thus, option b) is also correct.

c) $b \mid (a \cdot c)$

To check if $b \mid (a \cdot c)$, we know that $a \mid b$ and $a \mid c$, but we don't necessarily know that $b$ divides $a \cdot c$. In fact, we have no general guarantee that this is true, because $b$ might not divide $a \cdot c$ without further information about their relationships. So, this option is not necessarily true in general.

Thus, option c) is incorrect.

d) $c \mid (b + c)$

To check if $c \mid (b + c)$, we know that $a \mid b$ and $a \mid c$. However, for $c$ to divide $b + c$, we would need $b + c = c \cdot k$ for some integer $k$, which is not generally true. Therefore, there’s no guarantee that $c \mid (b + c)$ holds in general.

Thus, option d) is incorrect.

Conclusion:

• Correct options: a) and b)

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Would you like further clarification or additional details?

Related Questions:

1. What does divisibility mean in discrete mathematics?
2. Can you explain how to prove divisibility using integer representations?
3. How do products and sums behave under divisibility rules?
4. Is there a situation where option $c$ or $d$ could hold true?
5. How does the Euclidean algorithm help with understanding divisibility?

Tip: Divisibility problems can often be tackled by expressing numbers in terms of their divisors (e.g., $a \mid b \implies b = a \cdot k$). This simplifies proofs and helps in verifying relationships.