`2^(11) + 2^(12) + 2^(13) + 2^(14)` is divisible by _________.

Options

A. 7
B. 11
C. 3
D. 9

Answer

To solve the problem \(2^{11} + 2^{12} + 2^{13} + 2^{14}\) and determine what it is divisible by, we can follow these steps:

### Step 1: Factor out the common term
We notice that all terms in the expression have a common factor of \(2^{11}\). So, we can factor this out:

\[ 2^{11} + 2^{12} + 2^{13} + 2^{14} = 2^{11}(1 + 2 + 2^2 + 2^3) \]

### Step 2: Simplify the expression inside the parentheses
Now, we simplify the expression inside the parentheses:

\[ 1 + 2 + 2^2 + 2^3 = 1 + 2 + 4 + 8 \]

Calculating this gives:

\[ 1 + 2 = 3 \\ 3 + 4 = 7 \\ 7 + 8 = 15 \]

So, we have:

\[ 2^{11}(15) \]

### Step 3: Determine the divisibility of the product
Now we need to find out what \(2^{11} \times 15\) is divisible by.

- \(2^{11}\) is divisible by \(2\) and all powers of \(2\) up to \(2^{11}\).
- \(15\) can be factored into \(3 \times 5\).

Thus, \(2^{11} \times 15\) is divisible by \(2\), \(3\), \(5\), and \(15\).

### Step 4: Check the options for divisibility
The options given are \(7\), \(11\), \(3\), and \(9\).

- **Divisibility by \(7\)**: \(15\) is not divisible by \(7\).
- **Divisibility by \(11\)**: \(15\) is not divisible by \(11\).
- **Divisibility by \(3\)**: \(15\) is divisible by \(3\).
- **Divisibility by \(9\)**: \(15\) is not divisible by \(9\).

Thus, the only option that \(2^{11} + 2^{12} + 2^{13} + 2^{14}\) is divisible by is \(3\).

### Final Answer
The expression \(2^{11} + 2^{12} + 2^{13} + 2^{14}\) is divisible by **3**.

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