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?
  1. w + x + y + z = 13
  2. N + 5 is divisible by 9
Correct Answer: D
  1. 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:
    Figure 1
  2. 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)
  3. 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.
  4. 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.
  5. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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