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Difficulty: HardNumber Properties and Integer Constraints in Data Sufficiency

If kk is a real number, is kk an integer?

(1) k2−kk^2 - k is an integer.
(2) k3−kk^3 - k is an integer.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Neither statement alone is sufficient because non-integer real numbers can produce integer outputs for k2−kk^2 - k or k3−kk^3 - k. However, when both statements are combined, expressing k3−kk^3 - k as m(k+1)m(k + 1) proves that kk must be a rational number. For any rational number ab\frac{a}{b} in lowest terms, (ab)2−ab\left(\frac{a}{b}\right)^2 - \frac{a}{b} can only be an integer if the denominator b=1b = 1, which proves definitively that kk must be an integer.

Step-by-Step Solution

1
Evaluate Statement (1) independently.
Statement (1) is NOT sufficient.
We are given that k2−k=mk^2 - k = m for some integer mm. If m=1m = 1, then k2−k−1=0k^2 - k - 1 = 0, which gives k=1+52k = \frac{1 + \sqrt{5}}{2}. Here, kk is not an integer, but k2−k=1k^2 - k = 1 is an integer. Thus, Statement (1) alone does not determine whether kk is an integer.
2
Evaluate Statement (2) independently.
Statement (2) is NOT sufficient.
We are given that k3−k=nk^3 - k = n for some integer nn. If n=1n = 1, then k3−k−1=0k^3 - k - 1 = 0. By the Intermediate Value Theorem, there exists a real number k∈(1,2)k \in (1, 2) satisfying this equation. Since 1<k<21 < k < 2, kk cannot be an integer, yet k3−k=1k^3 - k = 1 is an integer. Thus, Statement (2) alone does not determine whether kk is an integer.
3
Evaluate Statements (1) and (2) combined.
The combined statements are SUFFICIENT.
Let k2−k=m∈Zk^2 - k = m \in \mathbb{Z} and k3−k=n∈Zk^3 - k = n \in \mathbb{Z}. Note that k3−k=k(k2−1)=k(k2−k+k−1)=k(m+k−1)=mk+k2−k=mk+m=m(k+1)k^3 - k = k(k^2 - 1) = k(k^2 - k + k - 1) = k(m + k - 1) = mk + k^2 - k = mk + m = m(k + 1). Thus, n=m(k+1)n = m(k + 1). If m=0m = 0, then k2−k=0  ⟹  k=0k^2 - k = 0 \implies k = 0 or k=1k = 1, both of which are integers. If m≠0m \neq 0, then mk=n−m  ⟹  k=n−mmmk = n - m \implies k = \frac{n - m}{m}. Since mm and nn are integers, kk must be a rational number. Let k=abk = \frac{a}{b} in reduced form with gcd⁡(a,b)=1\gcd(a, b) = 1 and b≥1b \ge 1. Substituting into k2−k=mk^2 - k = m yields a2−abb2=m  ⟹  a(a−b)=mb2\frac{a^2 - ab}{b^2} = m \implies a(a - b) = m b^2. This means b2b^2 must divide a(a−b)a(a - b). Since gcd⁡(a,b2)=1\gcd(a, b^2) = 1, b2b^2 must divide a−ba - b, which implies b=1b = 1. Therefore, k=ak = a, so kk MUST be an integer.

Key Concept

Testing implicit real number constraints vs integer constraints using polynomial reduction and rational root properties.
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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