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Difficulty: MediumNumber Properties and Integer Constraints in Data Sufficiency

If xx and yy are non-zero real numbers, is xy\frac{x}{y} an integer?

(1) x2+y2=5xyx^2 + y^2 = 5xy
(2) xx is a prime number and xyxy is an integer.

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) allows us to set up a quadratic equation for the ratio xy\frac{x}{y}, yielding (xy)2−5(xy)+1=0\left(\frac{x}{y}\right)^2 - 5\left(\frac{x}{y}\right) + 1 = 0. The roots of this quadratic equation are 5±212\frac{5 \pm \sqrt{21}}{2}, which are irrational numbers. Therefore, xy\frac{x}{y} cannot be an integer under any circumstance, providing a definitive 'No' answer to the question. Statement (1) is therefore sufficient. Statement (2) allows xy\frac{x}{y} to be an integer (e.g., x=3,y=1x=3, y=1) or a non-integer (e.g., x=3,y=2x=3, y=2), so it is not sufficient. Thus, the option stating that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient is the correct choice.

Step-by-Step Solution

1
Analyze the target question stem
The target asks if the quotient xy\frac{x}{y} is an integer, where xx and yy are real numbers (not restricted to integers).
Establishing that xx and yy are real numbers prevents making unjustified integer assumptions.
2
Evaluate Statement (1): x2+y2=5xyx^2 + y^2 = 5xy
Divide both sides by y2y^2 (since y≠0y \neq 0): (xy)2+1=5(xy)\left(\frac{x}{y}\right)^2 + 1 = 5\left(\frac{x}{y}\right), which rearranges to (xy)2−5(xy)+1=0\left(\frac{x}{y}\right)^2 - 5\left(\frac{x}{y}\right) + 1 = 0. Setting k=xyk = \frac{x}{y}, we get k2−5k+1=0k^2 - 5k + 1 = 0. Solving for kk using the quadratic formula gives k=5±212k = \frac{5 \pm \sqrt{21}}{2}.
Since 21\sqrt{21} is irrational, k=xyk = \frac{x}{y} is an irrational number and can NEVER be an integer.
3
Determine sufficiency for Statement (1)
Statement (1) yields a definitive 'No' to the question 'Is xy\frac{x}{y} an integer?'. Thus, Statement (1) alone is SUFFICIENT.
In Data Sufficiency Yes/No questions, a definitive 'No' answer is a sufficient result.
4
Evaluate Statement (2): xx is a prime number and xyxy is an integer
Case A: Let x=3x = 3 and y=1y = 1. Then xy=3xy = 3 (an integer), and xy=3\frac{x}{y} = 3 (an integer) -> YES.
Case B: Let x=3x = 3 and y=2y = 2. Then xy=6xy = 6 (an integer), and xy=32\frac{x}{y} = \frac{3}{2} (not an integer) -> NO.
Because xy\frac{x}{y} can be an integer or not an integer, Statement (2) alone is NOT sufficient.

Key Concept

Data Sufficiency Yes/No decision logic combined with irrational root analysis and real number constraints.
Estimated Time:2m 0s
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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