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Zorluk: ZorNumber Properties and Integer Constraints in Data Sufficiency

If nn is a positive integer, is nn divisible by 18?

(1) n2n^2 is divisible by 54.
(2) n3n^3 is divisible by 108.

  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.
Statement (1) establishes that n2n^2 is divisible by 54=21×3354 = 2^1 \times 3^3. Because the exponent of 3 in n2n^2 is 2b2b, the condition 2b≥32b \geq 3 implies b≥1.5b \geq 1.5, which forces b≥2b \geq 2 since bb must be an integer. Similarly, 2a≥1  ⟹  a≥12a \geq 1 \implies a \geq 1. Therefore, nn must contain at least 21×32=182^1 \times 3^2 = 18 as a factor, guaranteeing that nn is divisible by 18 (Definitive Yes). Statement (2) establishes that n3n^3 is divisible by 108=22×33108 = 2^2 \times 3^3, which only forces a≥1a \geq 1 and b≥1b \geq 1, so nn must only be a multiple of 6. Since n=6n = 6 yields a 'No' and n=18n = 18 yields a 'Yes', Statement (2) is insufficient.

Adım Adım Çözüm

1
Rephrase the question target in terms of prime factorization
For a positive integer nn to be divisible by 18=21×3218 = 2^1 \times 3^2, the prime factorization of nn must contain at least one factor of 2 and at least two factors of 3.
Decomposing the target integer into prime factors establishes clear minimum exponent thresholds for evaluation.
2
Evaluate Statement (1): n2n^2 is divisible by 54=21×3354 = 2^1 \times 3^3
If n=2a×3b×kn = 2^a \times 3^b \times k, then n2=22a×32b×k2n^2 = 2^{2a} \times 3^{2b} \times k^2. Divisibility by 21×332^1 \times 3^3 requires 2a≥1  ⟹  a≥12a \geq 1 \implies a \geq 1 and 2b≥3  ⟹  b≥22b \geq 3 \implies b \geq 2 (since aa and bb are integers). Thus nn must contain at least 21×32=182^1 \times 3^2 = 18, giving a definitive YES.
Integer constraints on exponents force fractional lower bounds to round up to the next integer.
3
Evaluate Statement (2): n3n^3 is divisible by 108=22×33108 = 2^2 \times 3^3
If n3=23a×33b×k3n^3 = 2^{3a} \times 3^{3b} \times k^3, then 3a≥2  ⟹  a≥13a \geq 2 \implies a \geq 1 and 3b≥3  ⟹  b≥13b \geq 3 \implies b \geq 1. Thus nn must be divisible by 21×31=62^1 \times 3^1 = 6. Testing values: if n=6n = 6, 63=2166^3 = 216 (divisible by 108), but 6 is NOT divisible by 18 (NO). If n=18n = 18, 183=583218^3 = 5832 (divisible by 108), and 18 IS divisible by 18 (YES). Statement (2) is INSUFFICIENT.
Testing specific values demonstrates that Statement (2) allows both 'Yes' and 'No' outcomes.

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Prime Factor Exponent Analysis under Integer Constraints in Data Sufficiency
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