Let's analyze each statement carefully to determine which is true:

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(a) A number $k$ divides the sum of three consecutive integers $n, n+1, n+2$ if and only if it divides the middle integer $n+1$.

• The sum of three consecutive integers is: $n + (n+1) + (n+2) = 3n + 3 = 3(n+1).$ This means the sum is always divisible by 3. Therefore, $k$ dividing $3(n+1)$ does not necessarily imply it divides $n+1$, because $k$ might include a factor of 3.
• This is false.

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(b) An integer $n$ is divisible by 6 if and only if it is divisible by 3.

• For a number to be divisible by 6, it must be divisible by both 2 and 3. Divisibility by 3 alone is not sufficient.
• This is false.

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(c) For all integers $a, b,$ and $c$, $a | (bc)$ if and only if $a | b$ and $a | c$.

• Counterexample: Let $a = 2$, $b = 4$, and $c = 3$. Here, $a | (bc) = 2 | 12$ (true), but $a \nmid 3$.
• This is false.

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(d) For all integers $a, b,$ and $c$, $a | (b+c)$ if and only if $a | b$ and $a | c$.

• If $a | b$ and $a | c$, then $b = ak_1$ and $c = ak_2$ for some integers $k_1$ and $k_2$. Adding them: $b + c = ak_1 + ak_2 = a(k_1 + k_2),$ so $a | (b+c)$.
• Conversely, $a | (b+c)$ does not imply $a | b$ and $a | c$, since a single $a$ dividing $b+c$ doesn't guarantee separate divisibility.
• This is false.

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(e) If $r$ and $s$ are integers, then $r | s$ if and only if $r^2 | s^2$.

• If $r | s$, then $s = rk$, so $s^2 = r^2k^2$, meaning $r^2 | s^2$.
• Conversely, if $r^2 | s^2$, then $s^2 = r^2k$. Since the square of a number is divisible by another square, $r$ must divide $s$.
• This is true.

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Answer: (e) is the correct statement.

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Do you want me to expand any explanation or clarify? Here are related questions to deepen your understanding:

1. What are the divisibility rules for integers?
2. Can you provide another counterexample for statement (c)?
3. How does the distributive property help prove statement (d)?
4. Why does statement (b) fail when $n$ is odd?
5. What other properties of divisors can we infer from statement (e)?

Tip: When analyzing "if and only if" statements, always check both directions to confirm equivalence.