Practice GMAT Data Sufficiency Question

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If x and y are positive integers, is the following cube root an integer?
Figure 1
  1. x = y2(y-1)
  2. x = 2
Correct Answer: A
  1. Evaluate Statement (1) alone.
    1. Substitute the value of x from Statement (1) into the equation and manipulate it algebraically.
      Figure 2
      Figure 4
    2. Since the question says that y is a positive integer, you know that the cube root of y3, which equals y, will also be a positive integer. Statement (1) is SUFFICIENT.
  2. Evaluate Statement (1) alone (Alternative Method).
    1. For the cube root of a number to be an integer, that number must be an integer cubed. Consequently, the simplified version of this question is: "is x + y2 equal to an integer cubed?"
    2. Statement (1) can be re-arranged as follows:
      x = y3 - y2
      y3 = x + y2
      Since y is an integer, the cube root of y3, which equals y, will also be an integer.
    3. Since y3 = x + y2, the cube root of x + y2 will also be an integer. Therefore, the following will always be an integer:
      Figure 1
    4. Statement (1) alone is SUFFICIENT.
  3. Evaluate Statement (2) alone.
    1. Statement (2) provides minimal information. The question can be written as: "is the following cube root an integer?"
      Figure 3
    2. If y = 4, x + y2 = 2 + 42 = 18 and the cube root of 18 is not an integer. However, if y = 5, x + y2 = 2 + 52 = 27 and the cube root of 27 is an integer. Statement (2) is NOT SUFFICIENT.
  4. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

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