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K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2

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BrentGMATPrepNow
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K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   12 Aug 2020, 09:32
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\(K = 10^{26} + 2^{26}\), and \(K\) is a multiple of \(2^n\) but NOT a multiple of \(2^{n+1}\). If n is a positive integer, what is the value of \(n+1\)?

A) 25
B) 26
C) 27
D) 28
E) 29
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D
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prabsahi
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   Updated on: 12 Aug 2020, 10:21
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Got stumped at first glance.So a good question for sure.

Here is my try:

K is a multiple of 2^N

So we can write

K.2^N = 2^26(5^26 +1)
SO,

2^N = 2^26 AND rest is expression for K
there N=26
now question asks N+1 .So 27 is my ans.

Correcting .I missed 5+1 will contribute another 2 here and so correct ans is
2^N= 2^27
implying N+1= 27
so N=28

Originally posted by prabsahi on 12 Aug 2020, 10:04.
Last edited by prabsahi on 12 Aug 2020, 10:21, edited 2 times in total.
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   12 Aug 2020, 10:17
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K.2^N = 2^26(5^26 +1)
now we now that 5^n nay number will end with 5 which is odd. when we add 1 to the 5 then last digit will always be 6.
which is divisible by 2 hence the whole expression is divisible by 2^27 also.
hence n+1 = 28
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   12 Aug 2020, 13:27
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BrentGMATPrepNow wrote:
\(K = 10^{26} + 2^{26}\), and \(K\) is a multiple of \(2^n\) but NOT a multiple of \(2^{n+1}\). If n is a positive integer, what is the value of \(n+1\)?

A) 25
B) 26
C) 27
D) 28
E) 29


IMPORTANT: later in my solution, we need to recognize that 5^b will end in 25 for all integer values of b greater than 0.
For example, 5^2 = 25
5^3 = 125
5^4 = 625
5^5 = 3125
5^6 = XXX25 etc....

So......
K = 10^26 + 2^26
= 2^26(5^26 + 1)
= 2^26(XXXX25 + 1) [aside: XXXX25 denotes some number ending in 25]
= 2^26(XXXX26)
= 2^26[2(XXXX3)] [Since XXX26 is EVEN, we can factor out a 2]
= 2^27[XXXX3]

Since XXXX3 is an ODD number, we can't factor out any more 2's.
This means K is a multiple of 2^27, but K is NOT a multiple 2^28.
In other words, n = 27

What is the value of n + 1?
Since n = 27, we can conclude that n + 1 = 27 + 1 = 28

Answer: D

Cheers,
Brent
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   12 Aug 2020, 22:56
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Hi Brent
Why does the answer say that it is C?

Posted from my mobile device
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   12 Aug 2020, 23:00
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Zuki1803 wrote:
Hi Brent
Why does the answer say that it is C?

Posted from my mobile device


I guess the answer needs correction.It will be D.
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   13 Aug 2020, 07:26
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Zuki1803 wrote:
Hi Brent
Why does the answer say that it is C?

Posted from my mobile device


The correct answer is D.
I don't see any indication that the answer was ever given as C (usually there's some notice stating that the question has been edited).
Are you sure it was C?
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   13 Aug 2020, 07:45
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Yeah,earlier it was C
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   13 Aug 2020, 10:07
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Zuki1803 wrote:
Yeah,earlier it was C


That's good to know, since I was thinking the % correct number was too high. Now that the correct answer is D, that number is decreasing (C will definitely be the most popular incorrect answer)
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   13 Sep 2021, 00:30
2^26(5^26+1)

Now 5^26+1 is divisible by 2

This is a multiple of 2^27

N+1 = 28

D
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink] New post   15 Apr 2024, 05:10
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Re: K = 10^26 + 2^26, and K is a multiple of 2^n but NOT a multiple of 2 [#permalink]
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