To determine if \( n^3 + 11 \) is divisible by 6 for any integer \( n \), we need to check if the expression \( n^3 + 11 \) is divisible by both 2 and 3.

### Step 1: Check for Divisibility by 2
An integer is divisible by 2 if it is even. The term \( n^3 \) will be:
- Even if \( n \) is even (since the cube of an even number is also even).
- Odd if \( n \) is odd (since the cube of an odd number is also odd).

Since \( 11 \) is an odd number:
- If \( n \) is even, \( n^3 \) is even, and \( n^3 + 11 \) will be odd (not divisible by 2).
- If \( n \) is odd, \( n^3 \) is odd, and \( n^3 + 11 \) will be even (divisible by 2).

Thus, \( n^3 + 11 \) is divisible by 2 only when \( n \) is odd.

### Step 2: Check for Divisibility by 3
We analyze \( n^3 \) modulo 3. There are three possible cases:
1. **If \( n \equiv 0 \pmod{3} \):** Then \( n^3 \equiv 0^3 \equiv 0 \pmod{3} \).
2. **If \( n \equiv 1 \pmod{3} \):** Then \( n^3 \equiv 1^3 \equiv 1 \pmod{3} \).
3. **If \( n \equiv 2 \pmod{3} \):** Then \( n^3 \equiv 2^3 \equiv 8 \equiv 2 \pmod{3} \).

For \( n^3 + 11 \) to be divisible by 3, \( n^3 \) must leave a remainder of 1 modulo 3, because \( 11 \equiv 2 \pmod{3} \) and \( 1 + 2 \equiv 0 \pmod{3} \). Thus, \( n \equiv 1 \pmod{3} \).

### Conclusion
For \( n^3 + 11 \) to be divisible by 6, \( n \) must be odd and satisfy \( n \equiv 1 \pmod{3} \).

In summary:
- \( n \) must be an odd integer.
- \( n \equiv 1 \pmod{3} \) (e.g., \( n = 1, 7, 13, \ldots \)).

Would you like further clarification on this?

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Here are five questions to deepen your understanding:
1. How can we determine if a number is divisible by any composite number, like 6 or 12?
2. What are the rules for determining divisibility by powers of primes, such as 8 or 9?
3. How does modular arithmetic help in proving divisibility properties?
4. Can the same approach be applied to other expressions involving \( n \), such as \( n^3 + 17 \)?
5. How does the parity (odd/even nature) of numbers influence divisibility?

**Tip:** Checking divisibility by composite numbers like 6 or 12 often involves verifying divisibility by their prime factors.

Explore how to check if the expression n^3 + 11 is divisible by 6 for any integer n using divisibility and modular arithmetic. Learn the rules and properties of divisibility by 2 and 3, with a step-by-step breakdown of number theory concepts.

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