`20^(2008)+16^(2008)-3^(2008)-1` is divisible by:

Options

A. 314
B. 323
C. 253
D. 91

Answer

To determine whether \(20^{2008} + 16^{2008} - 3^{2008} - 1\) is divisible by any of the given options, we can use the properties of modular arithmetic and the factorization of sums and differences of powers.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ E = 20^{2008} + 16^{2008} - 3^{2008} - 1 \]

2. **Group the Terms**:
We can rearrange the expression as follows:
\[ E = (20^{2008} - 3^{2008}) + (16^{2008} - 1) \]

3. **Apply the Difference of Powers**:
We can apply the difference of powers formula, which states that \(a^n - b^n\) is divisible by \(a - b\) for any integers \(a\), \(b\), and \(n\):
- For \(20^{2008} - 3^{2008}\), we have:
\[ 20^{2008} - 3^{2008} = (20 - 3)(20^{2007} + 20^{2006} \cdot 3 + \ldots + 3^{2007}) \]
This shows that \(E\) is divisible by \(20 - 3 = 17\).

- For \(16^{2008} - 1\), we can use:
\[ 16^{2008} - 1 = (16 - 1)(16^{2007} + 16^{2006} + \ldots + 1) \]
This shows that \(E\) is divisible by \(16 - 1 = 15\).

4. **Finding the Least Common Multiple**:
Now we need to find the least common multiple (LCM) of the divisors we found:
- The divisors are \(17\) and \(15\).
- The LCM of \(17\) and \(15\) is:
\[ \text{LCM}(17, 15) = 17 \times 15 = 255 \]

5. **Check Divisibility by the Options**:
Now we check which of the given options is divisible by \(255\):
- Option 1: \(114\) (not divisible)
- Option 2: \(323\) (not divisible)
- Option 3: \(253\) (not divisible)
- Option 4: \(91\) (not divisible)

However, we realize that \(255\) itself is not listed among the options. Therefore, we should check if \(17\) and \(15\) are factors of any of the options.

- \(323 = 17 \times 19\) (divisible by \(17\))
- \(15\) does not divide any of the options.

6. **Conclusion**:
Since \(E\) is divisible by \(17\) and \(323\) contains \(17\) as a factor, we conclude that the expression \(20^{2008} + 16^{2008} - 3^{2008} - 1\) is divisible by \(323\).

### Final Answer:
The expression \(20^{2008} + 16^{2008} - 3^{2008} - 1\) is divisible by **323**.