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Difficulty: MediumValue vs. Yes/No Data Sufficiency Decision Logic

If rr and ss are integers, is r+sr + s an even integer?

(1) r2−s2r^2 - s^2 is an odd integer.
(2) rr is an odd 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.
The correct response indicates that Statement (1) alone is sufficient while Statement (2) alone is not. Statement (1) establishes algebraically that r+sr + s must be odd because (r−s)(r+s)(r - s)(r + s) is an odd integer, yielding a definitive 'No' answer to whether r+sr + s is even. Statement (2) leaves the value and parity of ss completely unrestricted, allowing for both 'Yes' and 'No' outcomes.

Step-by-Step Solution

1
Rephrase the target question
The target is a Yes/No question: Is r+sr + s even? A statement is sufficient if it yields a definitive 'Yes' or a definitive 'No'.
In Yes/No Data Sufficiency questions, proving that a condition is always false (a definitive 'No') is just as sufficient as proving it is always true.
2
Evaluate Statement (1): r2−s2r^2 - s^2 is an odd integer
Factor r2−s2=(r−s)(r+s)r^2 - s^2 = (r - s)(r + s). Since the product of two integers is odd, both factors (r−s)(r - s) and (r+s)(r + s) must be odd integers. Thus, r+sr + s is definitely odd, which means r+sr + s is NOT even.
Since Statement (1) provides a definitive 'No' to the target question, Statement (1) alone is SUFFICIENT.
3
Evaluate Statement (2): rr is an odd integer
If r=3r = 3 and s=1s = 1, then r+s=4r + s = 4 (even →\rightarrow Yes). If r=3r = 3 and s=2s = 2, then r+s=5r + s = 5 (odd →\rightarrow No).
Because Statement (2) allows both 'Yes' and 'No' answers depending on the value of ss, Statement (2) alone is NOT sufficient.

Key Concept

Definitive Yes/No Decision Logic in Data Sufficiency
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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