Practice GMAT Data Sufficiency Question

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If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

w + x + y + z = 13
N + 5 is divisible by 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 order for a number, n, to be divisible by 9, its digits must add to nine. Likewise, the remainder of the sum of the digits of n divided by 9 is the remainder when n is divided by 9. In other words:
To see this, consider a few examples:
Let N = 901
901/9 = 100 + (R = 1)
(9+0+1)/9 = 10/9 = 1 + (R = 1)

Let N = 85
85/9 = 9 + (R = 4)
(8+5)/9 = 1 + (R = 4)

Let N = 66
66/9 = 7 + (R = 3)
(6+6)/9 = 1 + (R = 3)

Let N = 8956
8956/9 = 995 + (R = 1)
(8+9+5+6)/9 = 28/9 = 3 + (R = 1)
Evaluate Statement (1) alone.
1. Based upon what was shown above, since the sum of the digits of N is always 13, we know that remainder of N/9 will always be the remainder of 13/9, which is R=4.
2. In case this is hard to believe, consider the following examples:
  4540/9 = 504 + (R = 4)
  (4+5+4+0)/9 = 13/9 = 1 + (R = 4)
  
  1390/9 = 154 + (R = 4)
  (1+3+9+0)/9 = 13/9 = 1 + (R = 4)
  
  7231/9 = 803 + (R = 4)
  (7+2+3+1)/9 = 13/9 = 1 + (R = 4)
  
  1192/9 = 132 + (R = 4)
  (1+1+9+2)/9 = 13/9 = 1 + (R = 4)
3. Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
1. If adding 5 to a number makes it divisible by 9, there are 9-5=4 left over from the last clean division. In other words, N/9 will have a remainder of 4.
2. To help see this, consider the following examples:
  Let N = 4
  N+5=9 is divisible by 9 and N/9 -> R = 4
  
  Let N = 13
  N+5=18 is divisible by 9 and N/9 -> R = 4
  
  Let N = 724
  N+5=729 is divisible by 9 and N/9 -> R = 4
  
  Let N = 418
  N+5=423 is divisible by 9 and N/9 -> R = 4
3. Since N + 5 is divisible by 9, we know that the remainder of N/9 will always be 4. Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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