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Pls explain-Numper Property

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by theCodeToGMAT » Thu Oct 31, 2013 10:17 pm
Sum of digits of (10^x)^y - 64 is 279

10^xy - 64 will have digit sum as 279

If xy = 2 , result = 36, sum = 9
If xy = 3, result = 936, sum = 18
If xy = 4, result = 9936, sum = 27

So, AP,

(9) + (n-1)9 = 279

9 + 9n - 9 = 279

n = 31

So, n+1 = 32

Answer [spoiler]{E}[/spoiler]
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by mevicks » Thu Oct 31, 2013 10:25 pm
The sum of the digits of Image. What is the value of xy

A) 28
B) 29
C) 30
D) 31
E) 32
Image
Can be rewritten as: Image

So basically there will a pattern for the numbers:

xy = 2
100 - 64 = 36

xy = 3
1000 - 64 = 936

10000 - 64 = 9936
100000 - 64 = 99936...

So we need a xy from the answer choices which creates a sum of 270
The last two digits would be 36 (their sum is 9) so the remaining xy - 2 digits should add up to 279 - 9 = 270

Lets start with C:
xy = 30
28*9 < 270

xy = 31
29*9 < 270

xy = 32
30*9 = 270
STOP

[spoiler]Answer : E[/spoiler]
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by GMATGuruNY » Fri Nov 01, 2013 3:00 am
The sum of the digits of 10^(xy) - 64 is 279. What is the value of xy?
Test EASY CASES and look for a PATTERN.

10² - 64 = 36.
Sum of the digits = 3+6 = 9*1.

10³ - 64 = 936.
Sum of the digits = 9+3+6 = 9*2.

10� - 64 = 9936.
Sum of the digits = 9+9+3+6 = 9*3.

Notice the pattern:
The sum of the digits is of the form 9a, where the exponent in red -- the value of xy -- is equal to a+1.

For the integer in question, the sum of the digits = 279:
279 = 9a = 9*31.
Thus, the value of xy = a+1 = 31+1 = 32.

The correct answer is E.
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