Question

If x and y are positive integers, is x/y an integer?
1. Every factor of y is also a factor of x
2. Every factor of x is also a factor of y

• A. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
• B. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• C. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• D. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• E. EACH statement ALONE is sufficient to answer the question asked

Hint:

In factor-related Data Sufficiency questions, the most powerful tool is often the definition itself. The statement "Every factor of Y is a factor of X" directly implies that Y is a factor of X, which means X is a multiple of Y. Don't overcomplicate with prime factorizations unless necessary.

Answer 1

The Correct Option is D

Note: The question in the image is "is xy an integer?". Since x and y are defined as positive integers, their product xy is always an integer. This would make the question trivial. We assume the intended question, common in such exams, is "is x/y an integer?".

Step 1: Understanding the Concept:
The question asks whether \(x\) is a multiple of \(y\). This is a question about divisibility and factors. We need to analyze the conditions on the factors of \(x\) and \(y\) to determine if \(x/y\) is always an integer.

Step 2: Key Formula or Approach:
For \(x/y\) to be an integer, \(x\) must be a multiple of \(y\). This means that for the prime factorization of \(x = p_1^{a_1} p_2^{a_2} \dots\) and \(y = p_1^{b_1} p_2^{b_2} \dots\), we must have \(a_i \ge b_i\) for all prime factors \(p_i\).

Step 3: Detailed Explanation:
Analyzing Statement (1): Every factor of y is also a factor of x
Let's consider the number \(y\) itself. Since any number is a factor of itself, \(y\) must be a factor of \(x\).
If \(y\) is a factor of \(x\), it means that \(x\) is a multiple of \(y\).
This can be written as \(x = k \cdot y\) for some positive integer \(k\).
Then, the expression \(x/y\) becomes:
\[ \frac{x}{y} = \frac{k \cdot y}{y} = k \] Since \(k\) is an integer, \(x/y\) is an integer.
This statement guarantees that the answer to the question "is x/y an integer?" is always "Yes".
Therefore, Statement (1) ALONE is sufficient.
Analyzing Statement (2): Every factor of x is also a factor of y
By the same logic as above, this statement implies that \(x\) is a factor of \(y\).
This means that \(y\) is a multiple of \(x\).
This can be written as \(y = m \cdot x\) for some positive integer \(m\).
Now let's evaluate \(x/y\):
\[ \frac{x}{y} = \frac{x}{m \cdot x} = \frac{1}{m} \] The value \(\frac{1}{m}\) is an integer only if \(m=1\).
- If \(m=1\), then \(y=x\). In this case, \(x/y = 1\), which is an integer. (Answer: Yes) - If \(m=2\), then \(y=2x\). In this case, \(x/y = 1/2\), which is not an integer. (Answer: No)
Since we can get both a "Yes" and a "No" answer, this statement does not provide a definitive conclusion.
Therefore, Statement (2) ALONE is not sufficient.

Step 4: Final Answer:
Statement (1) is sufficient to answer the question, but Statement (2) is not. The correct option is (D).