Question

Is y an integer?
(1) 7y is an integer.
(2) y/7 is an integer.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

When testing number property statements, always try to prove them insufficient by finding counterexamples. If you can find one case that gives a "Yes" answer and another that gives a "No", the statement is insufficient. If all possible cases lead to the same answer, it is sufficient.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a Yes/No Data Sufficiency question about the properties of numbers. We need to determine if \(y\) must be an integer based on the given statements.
Step 2: Key Formula or Approach:
The definition of an integer is a whole number (not a fraction or decimal). We will test each statement by setting up an equation and seeing if \(y\) is forced to be an integer.
Step 3: Detailed Explanation:
Analyzing Statement (1):
"7y is an integer."
Let's express this mathematically. Let \(k\) be an integer.
\[ 7y = k \] Now, solve for \(y\):
\[ y = \frac{k}{7} \] Can \(y\) be a non-integer? Yes. For example, if we choose the integer \(k=1\), then \(y = 1/7\), which is not an integer. The answer to the question "Is y an integer?" is "No".
Can \(y\) be an integer? Yes. If we choose the integer \(k=14\), then \(y = 14/7 = 2\), which is an integer. The answer to the question "Is y an integer?" is "Yes".
Since we can get both a "Yes" and a "No" answer, statement (1) is not sufficient.
Analyzing Statement (2):
"y/7 is an integer."
Let's express this mathematically. Let \(m\) be an integer.
\[ \frac{y}{7} = m \] Now, solve for \(y\):
\[ y = 7m \] Since \(m\) is an integer, and the product of two integers (7 and \(m\)) is always an integer, \(y\) must be an integer.
This statement guarantees that \(y\) is an integer. The answer to the question "Is y an integer?" is always "Yes".
Therefore, statement (2) is sufficient.
Step 4: Final Answer:
Statement (2) alone is sufficient to answer the question, but statement (1) alone is not.