Step 1: Analyze the given information We are given that $x$, $y$, and $z$ are integers, and each is greater than $1$. We need to determine if $x$ is a prime number.

Step 2: Evaluate Statement I Statement I says: $xy^2 = 116$ Let us find the prime factorization of $116$: $116 = 2 \times 58 = 2 \times 2 \times 29 = 2^2 \times 29$ Since $y$ is an integer greater than $1$, $y^2$ must be a perfect square greater than $1$ that divides $116$. Looking at the prime factorization, the only perfect square factor of $116$ greater than $1$ is $2^2 = 4$. Therefore, $y^2 = 4$, which means $y = 2$. Substituting $y^2 = 4$ into the equation: $x \times 4 = 116$ $x = 29$ Since $29$ is a prime number, we can definitively answer "Yes" to the question. Thus, Statement I alone is sufficient.

Step 3: Evaluate Statement II Statement II says: $xz = 261$ Let us find the prime factorization of $261$: $261 = 3 \times 87 = 3 \times 3 \times 29 = 3^2 \times 29$ Since $x$ and $z$ are integers greater than $1$, $x$ can be any factor of $261$ other than $1$ and $261$. The possible values for $x$ are $3$, $9$, $29$, and $87$.

• If $x = 3$ or $x = 29$, then $x$ is a prime number.
• If $x = 9$ or $x = 87$, then $x$ is a composite number. Because $x$ can be either prime or composite, we cannot definitively answer the question. Thus, Statement II alone is not sufficient.

Conclusion The question can be answered using Statement I alone, but cannot be answered using Statement II alone. Therefore, the correct option is A.