Not clear :(

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The sum of the digits of [(10^x)^y]-64=279. What is the value of xy

A. 28
B. 29
C. 30
D. 31
E. 32


xy is positive integer

Ans-E

[[email protected]]

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

GMAT assassins aren't born, they're made,
Rich

[theCodeToGMAT]

[(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]

[kackerarnav]

[(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.

[theCodeToGMAT]

[(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

[Mathsbuddy]

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)

???

[[email protected]]

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.

GMAT assassins aren't born, they're made,
Rich

[Mathsbuddy]

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.

GMAT assassins aren't born, they're made,
Rich

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

[[email protected]]

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.

GMAT assassins aren't born, they're made,
Rich