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by [email protected] » Sun Nov 24, 2013 7:40 pm
Hi shibsriz,

This question has some awkward wording to it, but here's its intent: the number 10^(xy) - 64 has digits that add up to 279.

So, we need to figure out what THAT number is.

Here's what you need to "see" to solve this problem:

IF xy = 3, then 1,000 - 64 = 936 and the sum of digits = 18 = 2x9
If xy = 4, then 10,000 - 64 = 9,936 and the sum of digits = 27 = 3x9
If xy = 5, then 100,000 - 64 = 99,936 and the sum of digits = 36 = 4x9

Notice the pattern? The sum of digits is increasing by 9 every time.

So, how many times does 9 divide into 279? 31 times

As a reminder of the pattern:
xy = 3 --> 2x9
xy = 4 --> 3x9
xy = 5 --> 4x9

So, 31x9 --> xy = 32

Thus, the final answer is E

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by theCodeToGMAT » Sun Nov 24, 2013 9:59 pm
[(10^x)^y]-64=279

To find: xy

10^xy - 64 = 279

Let xy = 2 ==> 36 ==> Sum = 9
Let xy = 3 ==> 936 ==> Sum = 18
Let xy = 4 ==> 9936 ==> SUm = 27

So, using formula for AP

9 + (n-1)9 = 279
9n = 279
n = 31

So, 31+1 32

[spoiler]{E}[/spoiler]
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by kackerarnav » Mon Nov 25, 2013 8:00 am
theCodeToGMAT wrote:[(10^x)^y]-64=279

To find: xy

10^xy - 64 = 279

Let xy = 2 ==> 36 ==> Sum = 9
Let xy = 3 ==> 936 ==> Sum = 18
Let xy = 4 ==> 9936 ==> SUm = 27

So, using formula for AP

9 + (n-1)9 = 279
9n = 279
n = 31

So, 31+1 32

[spoiler]{E}[/spoiler]
Sorry, I was confused for a second because I didn't see a definition of n. :)
Last edited by kackerarnav on Mon Nov 25, 2013 8:03 am, edited 1 time in total.
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by theCodeToGMAT » Mon Nov 25, 2013 8:03 am
kackerarnav wrote:
theCodeToGMAT wrote:[(10^x)^y]-64=279

To find: xy

10^xy - 64 = 279

Let xy = 2 ==> 36 ==> Sum = 9
Let xy = 3 ==> 936 ==> Sum = 18
Let xy = 4 ==> 9936 ==> SUm = 27

So, using formula for AP

9 + (n-1)9 = 279
9n = 279
n = 31

So, 31+1 32

[spoiler]{E}[/spoiler]
Shouldn't that term in bold above be n-2, by the pattern?
No, I just used the formula for AP term==> a + (n-1)d

The part marked in green is what I did to cater that delta 1
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by Mathsbuddy » Mon Nov 25, 2013 8:13 am
10^xy - 64 -> 279

So 10^xy - 100 + 36 -> 279

So 10^xy - 100 -> 270

But 9 * 30 = 270 therefore there need to be 30 digits of 9

So 10^xy = 100 + 99999...00 (with 30 digits of 9)

So 10^xy = 10^31

Hence xy = 31 (D)

???
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by [email protected] » Mon Nov 25, 2013 2:45 pm
Hi Mathsbuddy,

You made a conceptual math error in your work. Remember that we have to add up the digits in the calculation. When you change the calculation (by turning - 64 into "-100 +36"), you're changing the digits. Unless you do the necessary steps to "undo" those changes later, you're going to end up with the wrong power of 10.

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by Mathsbuddy » Tue Nov 26, 2013 12:02 am
[email protected] wrote:Hi Mathsbuddy,

You made a conceptual math error in your work. Remember that we have to add up the digits in the calculation. When you change the calculation (by turning - 64 into "-100 +36"), you're changing the digits. Unless you do the necessary steps to "undo" those changes later, you're going to end up with the wrong power of 10.

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Edit

Thanks Rich,

Here's my corrected version (with some plagerism from your work!):

sum of digits of (10^xy - 64) -> 279

So sum of digits of (10^xy - 100 + 36) -> 279

So sum of digits of (10^xy - 100) -> 270

But 9 * 30 = 270 therefore there need to be 30 digits of 9

Note (QUOTED):
IF xy = 3, then 1,000 - 64 = 936 and the sum of digits = 18 = 2x9
If xy = 4, then 10,000 - 64 = 9,936 and the sum of digits = 27 = 3x9
If xy = 5, then 100,000 - 64 = 99,936 and the sum of digits = 36 = 4x9
So...
If xy = N, then 10^N - 64 = "(N-2) nines" + 36 and the sum of digits = (N-2) * 9
If xy = 32, then 10^32 - 64 = "30 nines" + 36 and the sum of digits = 270 = 31x9

So 10^xy = 10^32

Hence xy = 32
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by [email protected] » Tue Nov 26, 2013 1:24 am
Hi Mathsbuddy,

There you have it. The GMAT essentially "rewards" clever people (especially those who can "see" more than one way to solve a problem) with some easy-to-get points. However you choose to approach the Quant and Verbal sections, precision is a must, since the GMAT is also really good at punishing people who make silly mistakes. You're clearly a clever thinker, make sure that you keep your eye on the details.

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