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

If xx is a positive integer, is x2−1x^2 - 1 divisible by 8?

(1) xx is an odd integer.
(2) x=2kx = 2k for some integer kk.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  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. EACH statement ALONE is sufficient.Answer
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.
The correct choice indicates that each statement alone is sufficient. Statement (1) proves x2−1x^2 - 1 is always a multiple of 8 (a definitive 'Yes'), while Statement (2) proves x2−1x^2 - 1 is an odd integer and thus never a multiple of 8 (a definitive 'No'). Because both statements independently yield conclusive answers to the Yes/No question, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target algebraically
The target is whether x2−1=(x−1)(x+1)x^2 - 1 = (x - 1)(x + 1) is divisible by 8 for positive integer xx.
Factoring the expression reveals the product of consecutive even integers when xx is odd.
2
Evaluate Statement (1) independently
If xx is odd, let x=2m+1x = 2m + 1 for some non-negative integer mm. Then x2−1=(2m+1)2−1=4m2+4m=4m(m+1)x^2 - 1 = (2m + 1)^2 - 1 = 4m^2 + 4m = 4m(m + 1). Since m(m+1)m(m + 1) is the product of two consecutive integers, it is always even, so m(m+1)=2pm(m + 1) = 2p. Thus 4m(m+1)=8p4m(m + 1) = 8p, which is always divisible by 8. Statement (1) provides a definitive 'Yes'. Statement (1) is SUFFICIENT.
A statement that consistently yields a 'Yes' answer to a Yes/No DS question is sufficient.
3
Evaluate Statement (2) independently
If x=2kx = 2k for some integer kk, xx is an even integer. If xx is even, x2x^2 is even, which means x2−1x^2 - 1 is an odd integer. An odd integer can never be divisible by 8. Statement (2) provides a definitive 'No'. Statement (2) is SUFFICIENT.
In Yes/No Data Sufficiency, a statement that consistently yields a definitive 'No' is sufficient.
4
Synthesize the results from both statements
Since Statement (1) alone is sufficient and Statement (2) alone is sufficient, the correct option is the choice stating each statement alone is sufficient.
Both statements independently answer the stem question definitively.

Key Concept

In Yes/No Data Sufficiency, sufficiency requires a definitive, consistent answer ('Yes' for all cases OR 'No' for all cases). A statement yielding a consistent 'No' is fully sufficient.
Estimated Time:2m 0s
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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