`2^(25) + 2^(26) + 2^(27)` is divisible by :

Options

A. 9
B. 6
C. 5
D. 7

Answer

To solve the expression \(2^{25} + 2^{26} + 2^{27}\) and determine its divisibility, we can follow these steps:

### Step 1: Factor out the common term
We notice that all terms in the expression share a common factor of \(2^{25}\). Thus, we can factor it out:

\[ 2^{25} + 2^{26} + 2^{27} = 2^{25}(1 + 2 + 4) \]

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

\[ 1 + 2 + 4 = 7 \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression using our simplification:

\[ 2^{25} + 2^{26} + 2^{27} = 2^{25} \times 7 \]

### Step 4: Analyze the factors
The expression \(2^{25} \times 7\) consists of two factors: \(2^{25}\) and \(7\).

### Step 5: Determine divisibility
Now we can check the divisibility of the expression by the options provided:
- Since \(2^{25}\) is a power of 2, it is divisible by any power of 2.
- The factor \(7\) indicates that the entire expression is divisible by \(7\).

### Conclusion
Thus, the expression \(2^{25} + 2^{26} + 2^{27}\) is divisible by \(7\).

### Final Answer
The correct answer is option **B: 7**.

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