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Difficulty: HardNumber 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.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) 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.

Step-by-Step Solution

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.

Key Concept

Prime Factor Exponent Analysis under Integer Constraints in Data Sufficiency
Estimated Time:2m 0s
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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