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Difficulty: Very hardNumber Properties and Integer Constraints in Data Sufficiency

If pp and qq are positive integers, is p2+3qp^2 + 3q divisible by 6?

(1) p+qp + q is an odd integer.
(2) p2−q2p^2 - q^2 is divisible by 3.

  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 question requires determining whether p2+3qp^2 + 3q is divisible by 6, which demands divisibility by both 2 and 3. By parity rules, p2≡p(mod2)p^2 \equiv p \pmod 2 and 3q≡q(mod2)3q \equiv q \pmod 2, meaning p2+3qp^2 + 3q has the same parity as p+qp + q. Statement (1) specifies that p+qp + q is odd, so p2+3qp^2 + 3q is odd and cannot be divisible by 6. This gives a definitive 'No' answer, making Statement (1) alone sufficient. Statement (2) permits cases where p2+3qp^2 + 3q is divisible by 6 (e.g., p=3,q=3p=3, q=3) and cases where it is not (e.g., p=4,q=1p=4, q=1), making Statement (2) alone insufficient.

Step-by-Step Solution

1
Rephrase the question target using number properties and parity rules.
For p2+3qp^2 + 3q to be divisible by 6, it must be divisible by both 2 (even) and 3. For parity, p2≡p(mod2)p^2 \equiv p \pmod 2 and 3q≡q(mod2)3q \equiv q \pmod 2, so p2+3q≡p+q(mod2)p^2 + 3q \equiv p + q \pmod 2. Thus, p2+3qp^2 + 3q is even if and only if p+qp + q is even.
Simplifying the parity constraint simplifies evaluating statements regarding divisibility by 2.
2
Evaluate Statement (1): p+qp + q is an odd integer.
Since p+qp + q is odd, p2+3qp^2 + 3q must be odd. An odd integer cannot be divisible by 2, so p2+3qp^2 + 3q CANNOT be divisible by 6. This yields a definitive 'No'.
In Data Sufficiency Yes/No questions, a definitive 'No' answer means the statement is sufficient.
3
Evaluate Statement (2): p2−q2p^2 - q^2 is divisible by 3.
Test cases:
- If p=3,q=3p = 3, q = 3: p2−q2=0p^2 - q^2 = 0 (divisible by 3), and p2+3q=18p^2 + 3q = 18 (divisible by 6 -> YES).
- If p=4,q=1p = 4, q = 1: p2−q2=15p^2 - q^2 = 15 (divisible by 3), and p2+3q=19p^2 + 3q = 19 (not divisible by 6 -> NO).
Since Statement (2) yields both 'Yes' and 'No' outcomes, it is insufficient.

Key Concept

Parity rephrasing and Yes/No Data Sufficiency decision logic
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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