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Difficulty: MediumNumber Properties and Integer Constraints in Data Sufficiency

If xx is a positive integer, is x2+5xx^2 + 5x divisible by 6?

(1) xx is a prime number.
(2) x+1x + 1 is divisible by 3.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
  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 (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Factoring the stem expression gives x(x+5)x(x+5). For any positive integer xx, one of xx or x+5x+5 is even, making x(x+5)x(x+5) divisible by 2. Thus, the question reduces to whether x(x+5)x(x+5) is divisible by 3. Statement (1) allows x=2x=2 (x2+5x=14x^2+5x=14, not divisible by 6) and x=3x=3 (x2+5x=24x^2+5x=24, divisible by 6), yielding both 'No' and 'Yes' responses, so it is insufficient. Statement (2) states that x+1x+1 is a multiple of 3, meaning x≡2(mod3)x \equiv 2 \pmod 3. Evaluating x(x+5)≡2(2+5)=14≡2(mod3)x(x+5) \equiv 2(2+5) = 14 \equiv 2 \pmod 3 shows x2+5xx^2+5x is never divisible by 3, and therefore never divisible by 6. This consistent 'No' answer makes Statement (2) alone sufficient.

Step-by-Step Solution

1
Analyze and rephrase the question stem target expression.
Factor x2+5x=x(x+5)x^2 + 5x = x(x + 5). Since xx and x+5x + 5 have opposite parity, their product is always even (divisible by 2). Thus, x2+5xx^2 + 5x is divisible by 6 if and only if x(x+5)x(x + 5) is divisible by 3.
Determining divisibility by 6 requires checking divisibility by both 2 and 3.
2
Evaluate Statement (1): xx is a prime number.
If x=2x = 2, x2+5x=14x^2 + 5x = 14, which is not divisible by 6 (Answer: NO). If x=3x = 3, x2+5x=24x^2 + 5x = 24, which is divisible by 6 (Answer: YES). Since both 'Yes' and 'No' are possible, Statement (1) alone is INSUFFICIENT.
A statement must yield a single consistent answer ('Yes' or 'No') to be sufficient.
3
Evaluate Statement (2): x+1x + 1 is divisible by 3.
If x+1x + 1 is divisible by 3, then x=3k−1x = 3k - 1 for some integer k≥1k \ge 1. Substituting x=3k−1x = 3k - 1 into x(x+5)x(x + 5) yields (3k−1)(3k+4)=9k2+9k−4(3k - 1)(3k + 4) = 9k^2 + 9k - 4. Dividing this expression by 3 gives a remainder of 2. Because x(x+5)x(x + 5) is never divisible by 3, x2+5xx^2 + 5x is NEVER divisible by 6.
A definitive 'No' answer for all permitted values of xx establishes sufficiency.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency

Alternative Method

Test explicit values for Statement (2): x∈{2,5,8,11}x \in \{2, 5, 8, 11\}. Evaluating x2+5xx^2 + 5x yields 14,50,104,17614, 50, 104, 176, none of which are divisible by 6. Recognizing this uniform 'No' outcome confirms Statement (2) is sufficient.
Estimated Time:2m 0s
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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