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

Step 1: Understand what statement I tells us about x.
Statement I says x is divisible by 3 but not by 9. So x can be 3, 6, 12, 15, 21, 24, 30, 33, and so on, any multiple of 3 that is not also a multiple of 9.
This gives many possible values of x, not a single fixed value.

Step 2: Understand what statement II tells us about y.
Statement II says y is a multiple of 6. So y can be 6, 12, 18, 24, 30, and so on.
Again, this gives many possible values of y, not one fixed value.

Step 3: Combine both statements and test actual values.
Take x = 12 (divisible by 3, not by 9) and y = 6 (multiple of 6). Then \(x/y = 2\), which is a prime number.
Take x = 24 (divisible by 3, not by 9) and y = 6 (multiple of 6). Then \(x/y = 4\), which is not a prime number.
Take x = 6 (divisible by 3, not by 9) and y = 12 (multiple of 6). Then \(x/y = 0.5\), which is not even a whole number, so it cannot be prime.
So depending on which allowed values of x and y we pick, x/y is sometimes prime and sometimes not, and sometimes not even an integer.

Step 4: Conclusion.
Since both statements only restrict x and y to whole families of numbers, not to fixed values, the ratio x/y is never pinned down to one answer. Both statements together still leave the question undecided.
So the question cannot be answered even using both statements together. \[ \boxed{\text{Cannot be determined even with both statements}} \]