`(2^(71)+2^(72)+2^(73)+2^(74))` is divisible by

Options

A. 9
B. 10
C. 11
D. 13

Answer

To determine whether the expression \(2^{71} + 2^{72} + 2^{73} + 2^{74}\) is divisible by a certain number, we can simplify the expression step by step.

### Step-by-Step Solution:

1. **Factor out the common term:**
We notice that all terms in the expression have a common factor of \(2^{71}\). So, we can factor this out:
\[ 2^{71} + 2^{72} + 2^{73} + 2^{74} = 2^{71}(1 + 2 + 2^2 + 2^3) \]

2. **Simplify the expression inside the parentheses:**
Now, we need to simplify the expression inside the parentheses:
\[ 1 + 2 + 2^2 + 2^3 = 1 + 2 + 4 + 8 \]
Adding these together:
\[ 1 + 2 = 3 \\ 3 + 4 = 7 \\ 7 + 8 = 15 \]
Thus, we have:
\[ 1 + 2 + 2^2 + 2^3 = 15 \]

3. **Combine the results:**
Now we can rewrite our expression:
\[ 2^{71}(15) \]

4. **Check for divisibility:**
We need to check if \(2^{71} \times 15\) is divisible by a specific number. Since \(15 = 3 \times 5\), we can see that the expression is divisible by both \(3\) and \(5\).

5. **Conclusion:**
Therefore, \(2^{71} + 2^{72} + 2^{73} + 2^{74}\) is divisible by \(15\).