Practice GMAT Data Sufficiency Question

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If z = xⁿ - 19, is z divisible by 9?

x = 10; n is a positive integer
z + 981 is a multiple of 9

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: D

In working on this question, it is helpful to remember that a number will be divisible by 9 if the sum of its digits equals 9.
Evaluate Statement (1) alone.
1. Based upon the information in Statement (1), it is helpful to plug in a few values and see if a pattern emerges:
  10¹ - 19 = -9
  10² - 19 = 81; the sum of the digits is 9, which is divisible by 9, meaning the entire expression is divisible by 9
  10³ - 19 = 981; the sum of the digits is 9 + 8 + 1=18, which is divisible by 9, meaning the entire expression is divisible by 9
  10⁴ - 19 = 9981; the sum of the digits is 9(2) + 8 + 1=27, which is divisible by 9, meaning the entire expression is divisible by 9
2. Notice that, in each instance, the sum of the digits is divisible by 9, meaning the entire expression is divisible by 9.
3. The pattern that emerges is that there are (n-2) 9s followed by the digit 8 and the digit 1.
4. The pattern of the sum of the digits of 10ⁿ - 19 is 9(n-2) + 9 for all values of n > 1. (For n = 1, the sum is -9, which is also divisible by 9.) This means that the sum of the digits of 10ⁿ - 19 is 9(n-1). Since this sum will always be divisible by 9, the entire expression (i.e., 10ⁿ - 19) will always be divisible by 9.
5. Based upon this pattern, Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
1. Statement (2) says that z + 981 is a multiple of 9. This can be translated into algebra: 9(a constant integer) = z + 981
  Divide both sides by 9
2. Since 981 is divisible by 9 (its digits sum to 18, which is divisible by 9), you can further rewrite Statement (2).
3. Since an integer minus an integer is an integer, Statement (2) can be rewritten even further. Since z divided by 9 is an integer, z is divisible by 9. Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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