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

Options

A. 31
B. 13
C. 131
D. 33

Answer

To determine whether \( 16^5 + 2^{15} \) is divisible by any specific number, we can simplify the expression step by step.

### Step 1: Rewrite \( 16^5 \)
We know that \( 16 \) can be expressed as a power of \( 2 \):
\[ 16 = 2^4 \]
Thus, we can rewrite \( 16^5 \) as:
\[ 16^5 = (2^4)^5 \]
Using the power of a power property, this simplifies to:
\[ (2^4)^5 = 2^{4 \times 5} = 2^{20} \]

### Step 2: Rewrite the entire expression
Now we can rewrite the original expression \( 16^5 + 2^{15} \):
\[ 16^5 + 2^{15} = 2^{20} + 2^{15} \]

### Step 3: Factor out the common term
Next, we can factor out the common term \( 2^{15} \) from both terms:
\[ 2^{20} + 2^{15} = 2^{15}(2^{20-15} + 1) = 2^{15}(2^5 + 1) \]
Calculating \( 2^5 \):
\[ 2^5 = 32 \]
So, we have:
\[ 2^{15}(32 + 1) = 2^{15} \times 33 \]

### Step 4: Identify the divisibility
Now we see that the expression \( 16^5 + 2^{15} \) can be expressed as:
\[ 2^{15} \times 33 \]
This means that \( 16^5 + 2^{15} \) is divisible by both \( 2^{15} \) and \( 33 \).

### Conclusion
Thus, \( 16^5 + 2^{15} \) is divisible by \( 33 \).