Question

Directions: Each of the following questions is followed by two statements. Mark option (1) if the question can be answered using statement I alone. Mark option (2) if the question can be answered using statement II alone. Mark option (3) if both statements I and II together are needed to answer the question. Mark option (4) if the question cannot be answered even using both statements together.

Is x/y prime?

I. x is divisible by 3 but not by 9.
II. y is a multiple of 6.

• A. The question can be answered using statement I alone
• B. The question can be answered using statement II alone
• C. The question can be answered only if both statements I and II are used together
• D. The question cannot be answered even using both statements together

Hint:

Test a few actual number pairs that satisfy both clues; if x/y comes out prime for one valid pair and not prime for another, the data is not sufficient.

Answer 1

The Correct Option is D

We are asked whether $x/y$ is a prime number, and we are given two clues about $x$ and $y$ separately. To check sufficiency, the fastest method is to plug in a few different numbers that satisfy the clues and see if the answer to "is $x/y$ prime" stays the same every time.

• Clue about x: x is divisible by 3 but not by 9. This describes a whole family of numbers: 3, 6, 12, 15, 21, 24, 30, 33, and so on (skip the ones divisible by 9, like 9, 18, 27).
• Clue about y: y is a multiple of 6. This is also a family: 6, 12, 18, 24, 30, and so on.

1. Try x = 12, y = 6: both clues are satisfied (12 is divisible by 3 but not 9; 6 is a multiple of 6). Here $x/y = 2$, and 2 is prime.
2. Try x = 24, y = 6: both clues are satisfied again (24 is divisible by 3 but not 9; 6 is a multiple of 6). Here $x/y = 4$, and 4 is not prime.
3. Try x = 6, y = 12: both clues hold. Here $x/y = 0.5$, not even a whole number, so it cannot be prime.

Three valid pairs of x and y, all fitting both given clues, produce three different verdicts on whether $x/y$ is prime. That means the two statements, even combined, never pin down a single answer.

Let's summarize:

• Each clue describes a whole family of numbers, not one fixed value.
• Testing different members of these families gives different results for whether $x/y$ is prime.
• When the answer changes depending on which valid numbers we pick, the data is not sufficient.

So even using both statements together, we cannot say for certain whether $x/y$ is prime, and the correct option is (4).

\[ \boxed{\text{Answer: Not sufficient even with both statements}} \]