Question

Are \(x\) and \(y\) integers? Statement (I): Product \(xy\) is an integer.
Statement (II): \(x+y\) is an integer.

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

Whenever a statement seems sufficient, try constructing a counterexample. If one valid example gives integers and another valid example gives non-integers, the statement is not sufficient.

Answer 1

The Correct Option is D

Concept: In a Data Sufficiency question, a statement is sufficient only if it allows us to answer the question with certainty. The question asks whether both \(x\) and \(y\) are integers. To establish sufficiency, the statements must force \(x\) and \(y\) to be integers in every possible case.

Step 1: Analyze Statement (I) Alone. Statement (I) says: \[ xy=\text{an integer} \] This does not imply that \(x\) and \(y\) themselves are integers. Consider: \[ x=\frac12,\qquad y=4 \] Then \[ xy=\frac12\times 4=2 \] which is an integer. However, \[ x=\frac12 \] is not an integer. Therefore, Statement (I) alone is insufficient.

Step 2: Analyze Statement (II) Alone. Statement (II) says: \[ x+y=\text{an integer} \] Again, this does not guarantee that \(x\) and \(y\) are integers. For example, \[ x=\frac12,\qquad y=\frac12 \] Then \[ x+y=1 \] which is an integer. But neither \(x\) nor \(y\) is an integer. Hence Statement (II) alone is insufficient.

Step 3: Analyze Both Statements Together. Now we know: \[ xy=\text{integer} \] and \[ x+y=\text{integer} \] At first glance this may appear sufficient, but it is not. Case 1: Integer values possible Take \[ x=1,\qquad y=2 \] Then \[ xy=2 \] and \[ x+y=3 \] Both are integers, and \(x,y\) are integers. Case 2: Non-integer values also possible Take \[ x=\frac12,\qquad y=\frac32 \] Then \[ xy=\frac12\times\frac32=\frac34 \] which is not an integer. So choose another example: \[ x=\frac12,\qquad y=\frac52 \] Then \[ xy=\frac54 \] not integer. Consider \[ x=\frac12,\qquad y=\frac92 \] Again not integer. A better example is \[ x=\frac12,\qquad y=\frac32 \] which are roots of \[ t^2-2t+\frac34=0 \] Here, \[ x+y=2 \] (an integer) and \[ xy=\frac34 \] (not integer). Now take \[ x=\frac12,\qquad y=6 \] giving \[ xy=3 \] (integer) but \[ x+y=6.5 \] (not integer). To satisfy both simultaneously, consider \[ x=\frac12,\qquad y=\frac72 \] No. Instead, let \(x\) and \(y\) be roots of \[ t^2-2t+\frac34=0 \] which gives \[ x+y=2,\quad xy=\frac34. \] This shows that integer sum and integer product do not necessarily force integer roots in general. For example, \[ x=\frac{3+\sqrt5}{2},\qquad y=\frac{3-\sqrt5}{2} \] yield \[ x+y=3,\qquad xy=1 \] both integers, yet neither \(x\) nor \(y\) is an integer. Thus both statements together still do not determine whether \(x\) and \(y\) are integers. Therefore, the information remains insufficient. \[ \boxed{\text{Neither statement is sufficient}} \]