Practice GMAT Data Sufficiency Question
Return to the list of practice GMAT data sufficiency questions.
If z = xn - 19, is z divisible by 9?
- x = 10; n is a positive integer
- z + 981 is a multiple of 9
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.
- 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
- Notice that, in each instance, the sum of the digits is divisible by 9, meaning the entire expression is divisible by 9.
- The pattern that emerges is that there are (n-2) 9s followed by the digit 8 and the digit 1.
- 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.
- Based upon this pattern, Statement (1) is SUFFICIENT.
-
Evaluate Statement (2) alone.
- 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
- Since 981 is divisible by 9 (its digits sum to 18, which is divisible by 9), you can further rewrite Statement (2).
- 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.
Return to the list of practice GMAT data sufficiency questions.