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

Question

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

BrentGMATPrepNow · Accepted Answer

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 = 1 25 
5^4 = 6 25 
5^5 = 31 25  
5^6 = XXX 25  etc....

So......
K = 10^26 + 2^26
= 2^26(5^26 + 1)
= 2^26(XXXX 25  + 1)   25  denotes some number ending in  25 ] 
= 2^26(XXXX 26 )
= 2^26 2 (XXXX 3 )]   26  is EVEN, we can factor out a  2 ] 
=  2^27 3 ]

Since XXXX 3  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

prabsahi · Answer

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

Zuki1803 · Answer

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

rvgmat12 · Answer

2^26(5^26+1)

Now 5^26+1 is divisible by 2

This is a multiple of 2^27

N+1 = 28

D

vraju87 · Answer

Could anyone please explain how we can say with certainity that ( 5^{26} +1) will not have a power of 2 greater than 1. Thanks!

Bunuel · Answer

Good question.

First of all:

Divisibility rule for  2^n :

An integer is divisible by 2 if its last digit is divisible by 2.

An integer is divisible by 4 if its last two digits is divisible by 4.

An integer is divisible by 8 if its last three digits is divisible by 8.

...

Generally, an integer is divisible by  2^n  if its last  n  digits is divisible by  2^n .

Next, 5 raised to any positive integer power greater than 1 ends with 25. Thus, 5^26 + 1 = ...25 + 1 = ...26. Since 26 is not divisible by 4, then 5^26 + 1 is not divisible by any power of 2 greater than 1.

Hope it's clear.

vraju87 · Answer

Thank you for the clear explanation!

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards