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Difficulty: MediumNumber Properties and Integer Constraints in Data Sufficiency

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

(1) x2+2xx^2 + 2x is an integer.
(2) x3+2x2x^3 + 2x^2 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 to determine that xx is an integer, but neither statement alone is sufficient.
The correct response is that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) permits irrational values like x=−1+2x = -1 + \sqrt{2}, making it insufficient on its own. Statement (2) permits irrational values like x=−1+52x = \frac{-1 + \sqrt{5}}{2}, making it insufficient on its own. Combining both statements allows us to express xx as the quotient of two integers b/ab/a when x2+2x≠0x^2 + 2x \neq 0, which mathematically forces xx to be an integer.

Step-by-Step Solution

1
Evaluate Statement (1) independently: x2+2xx^2 + 2x is an integer.
Statement (1) is NOT sufficient.
Let x2+2x=1x^2 + 2x = 1. Solving x2+2x−1=0x^2 + 2x - 1 = 0 gives x=−1+2x = -1 + \sqrt{2}, which is a real non-integer. However, if x=1x = 1, x2+2x=3x^2 + 2x = 3 is also an integer. Since xx could be an integer or a non-integer, Statement (1) alone is insufficient.
2
Evaluate Statement (2) independently: x3+2x2x^3 + 2x^2 is an integer.
Statement (2) is NOT sufficient.
Let x3+2x2=1x^3 + 2x^2 = 1. The equation x3+2x2−1=0x^3 + 2x^2 - 1 = 0 factors as (x+1)(x2+x−1)=0(x + 1)(x^2 + x - 1) = 0. Setting x2+x−1=0x^2 + x - 1 = 0 yields non-integer root x=−1+52x = \frac{-1 + \sqrt{5}}{2}. For this non-integer value, x3+2x2=1x^3 + 2x^2 = 1, which is an integer. Since xx can also be the integer −1-1, Statement (2) alone is insufficient.
3
Evaluate Statements (1) and (2) together.
Both statements together are SUFFICIENT.
From Statement (1), let x2+2x=ax^2 + 2x = a, where aa is an integer. From Statement (2), let x3+2x2=bx^3 + 2x^2 = b, where bb is an integer. Notice that x3+2x2=x(x2+2x)=x⋅a=bx^3 + 2x^2 = x(x^2 + 2x) = x \cdot a = b. Case 1: If a=0a = 0, then x2+2x=0  ⟹  x(x+2)=0  ⟹  x=0x^2 + 2x = 0 \implies x(x + 2) = 0 \implies x = 0 or x=−2x = -2, both of which are integers. Case 2: If a≠0a \neq 0, then x=bax = \frac{b}{a}, meaning xx is a rational number. Let x=mnx = \frac{m}{n} in lowest terms where gcd⁡(m,n)=1\gcd(m, n) = 1 and n>0n > 0. Substituting x=mnx = \frac{m}{n} into x2+2x=ax^2 + 2x = a gives m2n2+2mn=a  ⟹  m2+2mn=an2  ⟹  m(m+2n)=an2\frac{m^2}{n^2} + \frac{2m}{n} = a \implies m^2 + 2mn = a n^2 \implies m(m + 2n) = a n^2. If n>1n > 1, any prime factor pp of nn must divide m(m+2n)m(m + 2n), which implies pp divides m2m^2 and thus pp divides mm. This contradicts gcd⁡(m,n)=1\gcd(m, n) = 1. Thus, nn must equal 11, proving xx is an integer.

Key Concept

Using algebraic combination of Data Sufficiency statements and integer polynomial constraints to establish sufficiency without assuming variables are integers.
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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