Practice GMAT Data Sufficiency Question

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If both x and y are positive integers less than 100 and greater than 10, is the sum x + y a multiple of 11?

x - y is a multiple of 22
The tens digit and the units digit of x are the same; the tens digit and the units digit of y are the same

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: B

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}
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.
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.
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.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

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