**Correct Answer: A**
**Step 1: Analyze Statement 1 alone**
Statement 1 states that $2^k - 1$ is a prime number. Numbers of the form $2^k - 1$ are known as Mersenne numbers. There is a fundamental mathematical property regarding these: for $2^k - 1$ to be prime, the exponent $k$ **must** itself be a prime number. $//$
Proof by contradiction: If $k$ were composite (e.g., $k = a \times b$), then $2^{ab} - 1$ would be divisible by $2^a - 1$ and $2^b - 1$. Since $2^k - 1$ is given as prime, $k$ cannot be composite. $//$
Wait, could $k = 1$? If $k = 1$, $2^1 - 1 = 1$, which is not prime. Since the statement says $2^k - 1$ **is** a prime number (the smallest being $2^2 - 1 = 3$), $k$ must be at least $2$. Thus, $k$ is definitely prime. Statement 1 is **sufficient**.
**Step 2: Analyze Statement 2 alone**
Statement 2 tells us $k$ is an odd number greater than $2$. $//$
If $k = 3$, it is prime. $//$
If $k = 9$, it is not prime (it is composite). $//$
Since $k$ can be either prime or composite, Statement 2 is **not sufficient**.
**Step 3: Analyze Statements 1 and 2 together**
Since Statement 1 is already sufficient to provide a "Yes" answer, we do not need to combine them, but they do not contradict each other.
**Logical Trap:**
The "Mersenne Prime" trap works in one direction only. While $k$ being prime is a **necessary** condition for $2^k - 1$ to be prime, it is not a **sufficient** condition. For example, if $k = 11$ (a prime), $2^{11} - 1 = 2047$, which is $23 \times 89$ (not prime). However, the question provides the primality of the result as a *fact* in Statement 1. Students often confuse the two directions of the logic. Additionally, students might overlook the $k=1$ case or fail to recognize the algebraic identity that forces $k$ to be prime.
**Conclusion:**
Statement 1 provides a definitive "Yes" based on number theory properties, while Statement 2 allows for both prime and composite values of $k$. Therefore, Statement 1 alone is sufficient.