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Difficulty: HardNumber Properties and Integer Constraints in Data Sufficiency

For a real number pp, is pp an integer?

(1) p3−pp^3 - p is a positive prime number.
(2) p2p^2 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) is sufficient because factoring p3−pp^3 - p gives (p−1)p(p+1)(p-1)p(p+1), which represents the product of three consecutive integers whenever pp is an integer. The product of any three consecutive integers must be divisible by 6. Since no prime number is a multiple of 6, pp cannot be an integer if p3−pp^3 - p is prime. This provides a definitive 'No' answer to the question 'Is pp an integer?', establishing sufficiency. Statement (2) is insufficient because pp could be an integer like 3 (yielding 'Yes') or an irrational number like 3\sqrt{3} (yielding 'No').

Step-by-Step Solution

1
Analyze Statement (1): p3−pp^3 - p is a positive prime number.
pp cannot be an integer, yielding a definitive 'No' to the question stem.
Factor p3−pp^3 - p as (p−1)p(p+1)(p-1)p(p+1). If pp were an integer, this product would represent three consecutive integers. Any three consecutive integers contain at least one even factor and exactly one multiple of 3, making their product divisible by 6. A prime number is a positive integer greater than 1 with no positive divisors other than 1 and itself, so no prime number can be a multiple of 6. Thus, no integer pp can make p3−pp^3 - p a prime number. Since p3−pp^3 - p is given to be a positive prime, pp must be a non-integer real number. A definitive 'No' answer establishes that Statement (1) alone is sufficient.
2
Analyze Statement (2): p2p^2 is an integer.
pp could be an integer or a non-integer, so the question cannot be answered uniquely.
If p=3p = 3, p2=9p^2 = 9 (an integer), and pp is an integer (Yes). If p=3p = \sqrt{3}, p2=3p^2 = 3 (an integer), but pp is not an integer (No). Because both 'Yes' and 'No' outcomes are possible, Statement (2) alone is not sufficient.
3
Determine the overall Data Sufficiency choice.
Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.
Statement (1) provides a conclusive answer on its own, whereas Statement (2) remains ambiguous.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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