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Math Expert V
Joined: 02 Sep 2009
Posts: 58443
k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 78% (01:50) correct 22% (02:09) wrong based on 228 sessions

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k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?

A. 2^n - 8
B. 2^n - 2
C. 2^n
D. 2^n + 4
E. 2^n + 5

Kudos for a correct solution.

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Posts: 155
Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Bunuel wrote:
k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?

A. 2^n - 8
B. 2^n - 2
C. 2^n
D. 2^n + 4
E. 2^n + 5

Kudos for a correct solution.

Since 2^n + 7 is divisible by 9,
2^n+7+9 and 2^n+7-9 should also be divisible by 9.
So our numbers are 2^n+16 and 2^n-2

General Discussion
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4009
Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Bunuel wrote:
k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?

A. 2^n - 8
B. 2^n - 2
C. 2^n
D. 2^n + 4
E. 2^n + 5

Kudos for a correct solution.

Kunal is using a nice rule that goes something like this:

Given: k, M and N are integers
If k is a divisor of both N and M, then k is a divisor of N+M (and N–M and M–N)

We're told that 9 is a divisor of 2^n + 7
We also know that 9 is a divisor of 9.
So, applying the above rule, 9 is a divisor of 2^n + 7 + 9, and 9 is a divisor of 2^n + 7 - 9

For more on this rule and other divisor rules, see our free video: http://www.gmatprepnow.com/module/gmat- ... /video/831

Cheers,
Brent
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Hi All,

This question can be solved with a bit of arithmetic and some 'brute force' (it also helps if you know the 'rule of 9' so that you can avoid having to do lots of division).

We're given a series of facts to work with:
1) K = (2^N) + 7 where N is an integer that is greater than 1.
2) K is divisible by 9

We're asked which of the 5 answer choices MUST also be divisible by 9.

**
Note: The 'rule of 9' can help you to quickly determine whether a number is divisible by 9 or not. To use this rule, you must add up the DIGITS of the number. If THAT total is divisible by 9, then the original number is divisible by 9.

For example:
18 is divisible by 9 because the digits "1" + "8" = 9, which is divisible by 9 --> thus, 18 IS divisible by 9
42 is NOT divisible by 9 because the digits "4" + "2" = 6, which is NOT divisible by 9 --> thus, 42 is NOT divisible by 9
**

If you don't see the 'pattern' involved in this question, then that's fine - you can still get to the solution using basic arithmetic. We just have to figure out the smallest value of N that fits everything that we've been told.

N = 2, so K = 4+7 = 11. 11 is NOT divisible by 9 though, so N CANNOT be 2
N = 3, so K = 15 --> NOT divisible by 9, so N CANNOT be 3
N = 4, so K = 23 --> NOT --> N CANNOT be 4
N = 5, so K = 39 --> NOT --> N CANNOT be 5
N = 6, so K = 71 --> NOT --> N CANNOT be 6

N = 7, so K = 135. 135 IS divisible by 9...so N CAN be 7

Let's take THAT value of N and use it against the answer choices. When you plug N=7 into the 5 choices, only one of them is divisible by 9...

GMAT assassins aren't born, they're made,
Rich
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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amanvermagmat wrote:
The question asks which of the following MUST be divisible by 9. So if we take any value of n that satisfies the question stem, it should also satisfy the correct answer option.

Now k = 2^n + 7, is divisible by 9. We can see if we put n=1, k = 2^1+7 = 9, which is divisible by 9. So if we put n=1, it satisfies the given condition. We have to now put n=1 in the answer options and see which one satisfies:

A) 2^1 - 8 = -6, NOT divisible by 9
B) 2^1 - 2 = 0, YES. This is divisible by 9
C) 2^1 = 2, NOT divisible by 9
D) 2^1 + 4 = 6, NOT divisible by 9
E) 2^1 + 5 = 7, NOT divisible by 9

Hi amanvermagmat,

While you did get the correct answer, it's worth noting that you did NOT follow the instructions in the prompt. We're told that N is an integer GREATER than 1, so using N=1 was technically NOT appropriate. By extension, in this question, you were fortunate to get the correct answer. On Test Day, GMAT question writers will often design at least one wrong answer that will 'punish' someone who is not following the 'rules' of the prompt (meaning that if you don't implement some essential piece of information, then you will end up thinking that one of the wrong answers is correct; this is especially true of DS questions). As such, you should be careful about how you approach questions in the future (and on Test Day).

GMAT assassins aren't born, they're made,
Rich
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Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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The question asks which of the following MUST be divisible by 9. So if we take any value of n that satisfies the question stem, it should also satisfy the correct answer option.

Now k = 2^n + 7, is divisible by 9. We can see if we put n=1, k = 2^1+7 = 9, which is divisible by 9. So if we put n=1, it satisfies the given condition. We have to now put n=1 in the answer options and see which one satisfies:

A) 2^1 - 8 = -6, NOT divisible by 9
B) 2^1 - 2 = 0, YES. This is divisible by 9
C) 2^1 = 2, NOT divisible by 9
D) 2^1 + 4 = 6, NOT divisible by 9
E) 2^1 + 5 = 7, NOT divisible by 9

Retired Moderator P
Joined: 22 Aug 2013
Posts: 1428
Location: India
Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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EMPOWERgmatRichC wrote:
amanvermagmat wrote:
The question asks which of the following MUST be divisible by 9. So if we take any value of n that satisfies the question stem, it should also satisfy the correct answer option.

Now k = 2^n + 7, is divisible by 9. We can see if we put n=1, k = 2^1+7 = 9, which is divisible by 9. So if we put n=1, it satisfies the given condition. We have to now put n=1 in the answer options and see which one satisfies:

A) 2^1 - 8 = -6, NOT divisible by 9
B) 2^1 - 2 = 0, YES. This is divisible by 9
C) 2^1 = 2, NOT divisible by 9
D) 2^1 + 4 = 6, NOT divisible by 9
E) 2^1 + 5 = 7, NOT divisible by 9

Hi amanvermagmat,

While you did get the correct answer, it's worth noting that you did NOT follow the instructions in the prompt. We're told that N is an integer GREATER than 1, so using N=1 was technically NOT appropriate. By extension, in this question, you were fortunate to get the correct answer. On Test Day, GMAT question writers will often design at least one wrong answer that will 'punish' someone who is not following the 'rules' of the prompt (meaning that if you don't implement some essential piece of information, then you will end up thinking that one of the wrong answers is correct; this is especially true of DS questions). As such, you should be careful about how you approach questions in the future (and on Test Day).

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

Hi

Yes, I missed it. Thank you for pointing that out. Kudos.
Intern  B
Joined: 21 Aug 2015
Posts: 22
Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Another approach can be

We know that k = 2^n + 7 is divisible by 9 so we can say that 2^n when divided by 9 is leaving a remainder 2 so definitely 2^n - 2 must be divisible by 9.

Please let me know if my approach is right.
Target Test Prep Representative G
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Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Bunuel wrote:
k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?

A. 2^n - 8
B. 2^n - 2
C. 2^n
D. 2^n + 4
E. 2^n + 5

Since k/9 = integer, (2^n + 7)/9 = integer. Since 2^n + 7 is a multiple of 9, we can add or subtract any multiple of 9 to 2^n + 7 and that expression will still be divisible by 9. Thus:

2^n + 7 - 9 = 2^n - 2 MUST be divisible by 9.

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k = 2^n + 7, where n is an integer greater than 1. If k is divisible  [#permalink]

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Bunuel wrote:
k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?

A. 2^n - 8
B. 2^n - 2
C. 2^n
D. 2^n + 4
E. 2^n + 5

Kudos for a correct solution.

I used brute force. Powers of 2 are easy to list, and all we need is a number + 7 that is divisible by 9. If number gets large, all we need is a number whose digits sum to 9.

Powers of 2 (must be greater than 1):
2$$^2$$, 2$$^3$$, 2$$^4$$, 2$$^5$$, 2$$^6$$, 2$$^7$$, 2$$^8$$ =
4, 8, 16, 32, 64, 128, 256

As I wrote each power down, I added 7 to see if the result would be divisible by 9, and hence would be k. I did the divisibility part in my head until I reached 128 + 7 = 135. But digits of 135 sum to 9; 135 is divisible by 9.

So 135 = 128 + 7, and
128 = 2$$^7$$, n = 7

Then I reasoned backward. All the answer choices have 2$$^n$$. n = 7. Starting point, therefore, for all answers is 128.

If 128 must have 7 added to it to be divisible by 9 ... Go the other way on the number line, from 128, to reach another multiple of 9. If 7 up from a number equals multiple of 9, then 2 down from a number will also be a multiple of 9.

"2 down" from 128 = 128 - 2. 128 - 2 = 126. Those digits also sum to 9, and per prompt, 126 must be divisible by 9.

128 - 2 = 2$$^n$$ - 2. Answer B
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