Is divisible by -
`(2^(71)+2^72+2^73+2^74)`

Options

A. 9
B. 10
C. 11
D. 13

Answer

To determine if \(2^{71} + 2^{72} + 2^{73} + 2^{74}\) is divisible by \(2^7\), we can follow these steps:

### Step 1: Factor out the smallest power of 2
We notice that all terms in the expression share a common factor of \(2^{71}\). We can factor this out:
\[ 2^{71} + 2^{72} + 2^{73} + 2^{74} = 2^{71}(1 + 2 + 2^2 + 2^3) \]

### 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:
\[ 1 + 2 = 3 \\ 3 + 4 = 7 \\ 7 + 8 = 15 \]
Thus, we have:
\[ 2^{71}(15) \]

### Step 3: Check divisibility by \(2^7\)
Now, we need to check if \(2^{71}(15)\) is divisible by \(2^7\). Since \(2^{71}\) is a power of 2, it is certainly divisible by \(2^7\) because \(71 > 7\).

### Conclusion
Since \(2^{71}\) is divisible by \(2^7\), we conclude that the entire expression \(2^{71} + 2^{72} + 2^{73} + 2^{74}\) is divisible by \(2^7\).

### Final Answer
Yes, \(2^{71} + 2^{72} + 2^{73} + 2^{74}\) is divisible by \(2^7\).

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