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rss GMAT Practice Question (One of Hundreds)
If both x and y are positive integers less than 100 and greater than 10, is the sum x + y a multiple of 11?
  1. x - y is a multiple of 22
  2. The tens digit and the units digit of x are the same; the tens digit and the units digit of y are the same
Correct Answer: B
If both x and y are multiples of 11, then both x + y and x - y will be multiples of 11.
  1. If both x and y are multiples of 11, then both x + y and x - y will be multiples of 11. In other words, if two numbers have a common divisor, their sum and difference retain that divisor.
    In case this is hard to conceptualize, consider the following examples:
    42 - 18 {both numbers share a common factor of 6}
    =(6*7) - (6*3)
    =6(7 - 3)
    =6(4)
    =24 {which is a multiple of 6}

    49 + 14 {both numbers share a common factor of 7}
    =(7*7) + (7*2)
    =7(7+2)
    =7*9
    =63 {which is a multiple of 7}
  2. However, if x and y are not both multiples of 11, it is possible that x - y is a multiple of 11 while x + y is not a multiple of 11. For example:
    68 - 46 = 22 but 68 + 46 = 114, which is not divisible by 11.
    The reason x - y is a multiple of 11 but not x + y is that, in this case, x and y are not individually multiples of 11.
  3. Evaluate Statement (1) alone.
    1. Since x-y is a multiple of 22, x-y is a multiple of 11 and of 2 because 22=11*2
    2. If both x and y are multiples of 11, the sum x + y will also be a multiple of 11. Consider the following examples:
      44 - 22 = 22 {which is a multiple of 11 and of 22}
      44 + 22 = 66 {which is a multiple of 11 and of 22}

      88 - 66 = 22 {which is a multiple of 11 and of 22}
      88 + 66 = 154 {which is a multiple of 11 and of 22}
    3. However, if x and y are not individually divisible by 11, it is possible that x - y is a multiple of 22 (and 11) while x + y is not a multiple of 11. For example:
      78 - 56 = 22 but 78 + 56 = 134 is not a multiple of 11.
    4. Statement (1) alone is NOT SUFFICIENT.
  4. Evaluate Statement (2) alone.
    1. Since the tens digit and the units digit of x are the same, the range of possible values for x includes:
      11, 22, 33, 44, 55, 66, 77, 88, 99
      Since each of these values is a multiple of 11, x must be a multiple of 11.
    2. Since the tens digit and the units digit of y are the same, the range of possible values for y includes:
      11, 22, 33, 44, 55, 66, 77, 88, 99
      Since each of these values is a multiple of 11, y must be a multiple of 11.
    3. As demonstrated above, if both x and y are a multiple of 11, we know that both x + y and x - y will be a multiple of 11.
    4. Statement (2) alone is SUFFICIENT.
  5. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.
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