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

Options

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

Answer

To solve the problem \( 16^5 + 2^{15} \) and determine what it is divisible by, we can break it down step by step.

### Step 1: Rewrite \( 16^5 \) in terms of base 2
We know that \( 16 \) can be expressed as \( 2^4 \). Therefore, we can rewrite \( 16^5 \) as:
\[ 16^5 = (2^4)^5 = 2^{4 \times 5} = 2^{20} \]

### Step 2: Rewrite the expression
Now, substituting \( 16^5 \) back into the original expression, we have:
\[ 16^5 + 2^{15} = 2^{20} + 2^{15} \]

### Step 3: Factor out the common term
Both terms in the expression \( 2^{20} + 2^{15} \) have a common factor of \( 2^{15} \). We can factor this out:
\[ 2^{20} + 2^{15} = 2^{15}(2^{5} + 1) \]

### Step 4: Simplify the expression inside the parentheses
Now we simplify \( 2^5 + 1 \):
\[ 2^5 = 32 \quad \text{so} \quad 2^5 + 1 = 32 + 1 = 33 \]

### Step 5: Write the final expression
Now we can write the entire expression as:
\[ 2^{15}(33) \]

### Step 6: Determine the divisibility
Since the expression is \( 2^{15} \times 33 \), it is clear that \( 16^5 + 2^{15} \) is divisible by \( 33 \).

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

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