`16^(5)+2^(15)` is divisible by

Options

A. 31
B. 13
C. 27
D. 33

Answer

To solve the expression \(16^{5} + 2^{15}\) and determine its divisibility, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 16^{5} + 2^{15} \]
We can rewrite \(16\) as \(2^{4}\):
\[ (2^{4})^{5} + 2^{15} \]

### Step 2: Simplify using the power of a power rule
Using the power of a power rule \((a^{m})^{n} = a^{m \cdot n}\), we simplify:
\[ 2^{4 \cdot 5} + 2^{15} = 2^{20} + 2^{15} \]

### Step 3: Factor out the common term
Next, we can factor out the common term \(2^{15}\):
\[ 2^{15}(2^{5} + 1) \]

### Step 4: Calculate \(2^{5} + 1\)
Now, we calculate \(2^{5} + 1\):
\[ 2^{5} = 32 \quad \text{so} \quad 32 + 1 = 33 \]
Thus, we have:
\[ 2^{15} \cdot 33 \]

### Step 5: Conclusion about divisibility
Since the expression can be written as \(2^{15} \cdot 33\), it is clear that the entire expression \(16^{5} + 2^{15}\) is divisible by \(33\).

### Final Answer
The expression \(16^{5} + 2^{15}\) is divisible by \(33\).

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