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Difficulty: EasyNumber Properties and Integer Constraints in Data Sufficiency

Is xx an integer?

(1) 3x3x is an integer.
(2) 5x5x 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.
Evaluating Statement (1) alone shows xx could be 13\frac{1}{3} (not an integer) or 11 (an integer), so Statement (1) is insufficient. Evaluating Statement (2) alone shows xx could be 15\frac{1}{5} (not an integer) or 11 (an integer), so Statement (2) is insufficient. Taking both statements together, 3x=a3x = a and 5x=b5x = b for integers aa and bb. Subtracting 5x5x from 2(3x)2(3x) yields 6x−5x=x=2a−b6x - 5x = x = 2a - b. Because integers are closed under multiplication and subtraction, 2a−b2a - b must be an integer, confirming that xx is definitively an integer.

Step-by-Step Solution

1
Evaluate Statement (1) independently
If 3x=a3x = a where aa is an integer, then x=a3x = \frac{a}{3}. If a=1a = 1, x=13x = \frac{1}{3} (not an integer). If a=3a = 3, x=1x = 1 (an integer).
Since xx can be either an integer or a non-integer, Statement (1) alone is not sufficient.
2
Evaluate Statement (2) independently
If 5x=b5x = b where bb is an integer, then x=b5x = \frac{b}{5}. If b=1b = 1, x=15x = \frac{1}{5} (not an integer). If b=5b = 5, x=1x = 1 (an integer).
Since xx can be either an integer or a non-integer, Statement (2) alone is not sufficient.
3
Evaluate Statements (1) and (2) together
Since 3x3x and 5x5x are both integers, their linear combination 2(3x)−5x=6x−5x=x2(3x) - 5x = 6x - 5x = x must also be an integer.
The difference between two integers is always an integer, proving conclusively that xx is an integer.

Key Concept

Linear combinations of real numbers and integer constraints in Data Sufficiency
Estimated Time:1m 0s
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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