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Data Sufficiency 1 / 8

Is the integer $k$ a prime number?
Statement 1: $2^k - 1$ is a prime number.
Statement 2: $k$ is an odd number such that $3 < k < 10$.
Options:
Choose option A: If Statement 1 ALONE is sufficient to answer the question.
Choose option B: If Statement 2 ALONE is sufficient to answer the question.
Choose option C: If Statements 1 & 2 TOGETHER are sufficient but are not sufficient independently.
Choose option D: If Each statement is INDEPENDENTLY sufficient.
Choose option E: If Statements together are NOT sufficient. Additional data is required.

Correct Answer: A
Step 1: Analyze Statement 1 alone
Statement 1 states that $2^k - 1$ is a prime number (known as a Mersenne prime). A fundamental property of Mersenne primes is that for $2^k - 1$ to be prime, the exponent $k$ must itself be a prime number.
Proof by contrapositive: If $k$ is composite (e.g., $k = ab$), then $2^{ab} - 1$ is always divisible by $2^a - 1$ and $2^b - 1$.
Since $2^k - 1$ is prime, $k$ cannot be composite.
Note: $k$ cannot be $1$ because $2^1 - 1 = 1$, which is not prime. Therefore, $k$ must be a prime number ($2, 3, 5, 7, \dots$).
This statement consistently yields a "Yes" to the question. Statement 1 is sufficient.
Step 2: Analyze Statement 2 alone
Statement 2 states that $k$ is an odd number and $3 < k < 10$. The possible values for $k$ are:$k = 5$ (a prime number)$k = 7$ (a prime number)$k = 9$ (a composite number)
Since $k$ could be prime ($5, 7$) or composite ($9$), we cannot definitively answer if $k$ is prime. Statement 2 is not sufficient.
Step 3: Analyze Statements 1 and 2 together
Since Statement 1 is already sufficient independently, we do not need to combine them. However, if we did, the only possible values for $k$ would be $5$ and $7$ (since $2^9 - 1 = 511$, which is $7 \times 73$ and thus not prime). This would still lead to a "Yes," but it does not change the sufficiency of Statement 1.
Logical Trap: The most common trap here is twofold:
The Mersenne Fallacy: Students often forget the mathematical necessity that $k$ must be prime for $2^k - 1$ to be prime.
Number Constraints: In Statement 2, students frequently assume that all odd numbers are prime, overlooking $k = 9$.
Additionally, some might mistakenly think Statement 1 is only true if $k$ is an "odd prime," but the only even prime is $2$, and $2^2 - 1 = 3$ (which is prime), so $k = 2$ also satisfies Statement 1 and is a prime number.
Conclusion: Statement 1 provides a definitive mathematical condition that forces $k$ to be prime. Statement 2 leaves room for $k$ to be the composite number $9$. Thus, only Statement 1 is sufficient.

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