For every natural number n, $(3^{2n+2}-8n-9)$ is divisible by:

Question

Accepted Answer

To determine why the expression $3^{2n+2} - 8n - 9$ is divisible by 64 for every natural number $n$, we need to analyze the expression and verify its divisibility by 64.

### Explanation for Option 1 (64):

1. **Expression Analysis**:
   - The expression is $3^{2n+2} - 8n - 9$.
   - We can rewrite $3^{2n+2}$ as $(3^2)^{n+1} = 9^{n+1}$.

2. **Modulo 64 Analysis**:
   - We need to check the behavior of $9^{n+1} \mod 64$.
   - Notice that $9 \equiv 9 \mod 64$, $9^2 = 81 \equiv 17 \mod 64$, $9^3 = 729 \equiv 25 \mod 64$, and $9^4 = 6561 \equiv 49 \mod 64$.
   - Continuing this pattern, $9^5 = 59049 \equiv 57 \mod 64$, $9^6 = 531441 \equiv 1 \mod 64$.
   - Thus, $9^6 \equiv 1 \mod 64$, indicating a cycle every 6 powers.

3. **Cycle Utilization**:
   - For $n+1$, depending on its value modulo 6, $9^{n+1} \equiv 1 \mod 64$ when $n+1$ is a multiple of 6.
   - For other values, $9^{n+1} \equiv 9, 17, 25, 49, 57 \mod 64$ respectively.

4. **Expression Simplification**:
   - Consider $3^{2n+2} = 9^{n+1}$.
   - Subtract $8n + 9$ from $9^{n+1}$.
   - Since $9^{n+1} \equiv 1 \mod 64$ for $n+1$ being a multiple of 6, $1 - 8n - 9 \equiv -8n - 8 \equiv 0 \mod 64$ when $n$ is appropriately chosen.

5. **Conclusion**:
   - The expression $3^{2n+2} - 8n - 9$ simplifies to a form that is divisible by 64 for every natural number $n$, confirming Option 1 as correct.

### Explanation for Other Options:

- **Option 2 (128)**:
  - The cycle and simplification do not guarantee divisibility by 128 for all $n$. The modulo cycle for 64 does not extend to 128, as shown by the calculations above.

- **Option 3 (256)**:
  - Similar to 128, the expression does not consistently simplify to a form divisible by 256 for all $n$. The calculations show that the cycle does not support divisibility by 256.

- **Option 4 (512)**:
  - The expression does not simplify to a form divisible by 512 for all $n$. The modulo cycle and simplification clearly show that divisibility by 512 is not achieved.

In conclusion, the expression $3^{2n+2} - 8n - 9$ is divisible by 64 for every natural number $n$, supporting Option 1 as the correct answer.