Directions: Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly.

The correct option is (a) - Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement.

Analysis

Statement I alone: $x{y}^2=116$
We are given that x, y, and z are integers greater than 1. Let's find the prime factorization of 116:
$116=2\times 58=2\times 2\times 29=29\times 2^2$.
Since the equation is $x\times y^2=116$ and y must be an integer greater than 1, the only possible value for $y^2$ is $2^2$ (which is 4).
If $y^2=4$, then $y=2$ (which is > 1).
Substituting this back: $x\times 4=116$, which gives $x=29$.
Since 29 is a prime number, we get a definitive "Yes" to the question "Is x a prime number?".
Statement I alone is sufficient.

Statement II alone: $xz=261$
We are given $xz=261$, where x and z are integers greater than 1. Let's find the factors of 261:
$261=3\times 87=3\times 3\times 29=9\times 29$.
Possible pairs for (x, z) such that both are greater than 1 are:
1. $x=3$, $z=87$ (x is prime)
2. $x=9$, $z=29$ (x is NOT prime)
3. $x=29$, $z=9$ (x is prime)
4. $x=87$, $z=3$ (x is NOT prime)
Because x can be either prime (3, 29) or composite (9, 87), we cannot determine if x is definitely a prime number.
Statement II alone is not sufficient.

Why the other options are incorrect

• Option (b) - either statement alone: This is incorrect because Statement II fails to provide a unique answer (x could be 9 or 29), whereas Statement I provides a unique prime value for x.
• Option (c) - both statements together: This is incorrect because Statement I is already sufficient on its own. The "together" option is only chosen when neither statement works independently.
• Option (d) - cannot be answered: This is incorrect because Statement I provides a definitive mathematical proof that x must be 29, which is a prime number.

Key Concept

Prime factorization and integer constraints (x, y > 1) in algebraic equations to determine unique values.