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Zorluk: OrtaNumber Properties and Integer Constraints in Data Sufficiency

If nn is a positive integer, is n2+3n+2n^2 + 3n + 2 divisible by 12?

(1) nn is a multiple of 3.
(2) n+1n + 1 is a prime number.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
  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.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Factoring the stem expression gives n2+3n+2=(n+1)(n+2)n^2 + 3n + 2 = (n + 1)(n + 2). Statement (1) tells us nn is a multiple of 3, which implies neither n+1n + 1 nor n+2n + 2 is a multiple of 3. Consequently, their product can never be divisible by 3, and thus can never be divisible by 12. Because Statement (1) conclusively answers 'No' to the question stem, Statement (1) ALONE is sufficient. Statement (2) allows n=2n = 2 (giving 12, divisible by 12) and n=4n = 4 (giving 30, not divisible by 12), yielding both 'Yes' and 'No' responses, making Statement (2) alone insufficient.

Adım Adım Çözüm

1
Rephrase the question stem target using factoring.
The expression n2+3n+2n^2 + 3n + 2 factors into (n+1)(n+2)(n + 1)(n + 2). The question asks whether (n+1)(n+2)(n + 1)(n + 2) is divisible by 12=22×312 = 2^2 \times 3.
Factoring quadratic expressions helps reveal divisibility properties of consecutive integers.
2
Evaluate Statement (1): nn is a multiple of 3.
If n=3kn = 3k for some positive integer kk, then n+1=3k+1n + 1 = 3k + 1 and n+2=3k+2n + 2 = 3k + 2. Neither factor contains 3 as a prime factor, so their product (3k+1)(3k+2)(3k + 1)(3k + 2) leaves a remainder of 1×2=21 \times 2 = 2 when divided by 3. Since the product is never divisible by 3, it can NEVER be divisible by 12.
In a Yes/No Data Sufficiency question, a statement that yields a definitive 'No' to the question is SUFFICIENT.
3
Evaluate Statement (2): n+1n + 1 is a prime number.
If n=2n = 2, then n+1=3n + 1 = 3 (prime). The expression (2+1)(2+2)=12(2+1)(2+2) = 12, which is divisible by 12 (Answer: YES). If n=4n = 4, then n+1=5n + 1 = 5 (prime). The expression (4+1)(4+2)=30(4+1)(4+2) = 30, which is not divisible by 12 (Answer: NO). Since both 'Yes' and 'No' are possible, Statement (2) is NOT sufficient.
Testing specific values shows that Statement (2) does not yield a consistent answer.

Anahtar Kavram

Definitive Yes/No decision logic in Data Sufficiency combined with divisibility and prime factor properties of consecutive integers.
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Number Properties and Integer Constraints in Data Sufficiency

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