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Divisibilty

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Source: — Problem Solving |
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Re: Divisibilty

by tohellandback » Sun Aug 16, 2009 7:22 pm
clar wrote:If (2^18-n) is divisible by 3 which of the following could be the value of n?

1- 4
2- 5
3- 7

a) 1 only
b) 2 only
c) 3 only
d) 1 and 2
e) 2 and 3

Can someone pls help with this question?
2^18-n is divisible by 3, so 2^18 when divided by 3 gives remainder n

2^18 when divided by 3 gives remainder 1.( I hope you can figure this one out)

n=1,4,7,10,...
answer from the choices is 4 and 7.
but your options are wrong
n can be both 4 and 7.
Last edited by tohellandback on Sun Aug 16, 2009 7:36 pm, edited 1 time in total.
The powers of two are bloody impolite!!
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by clar » Sun Aug 16, 2009 7:35 pm
yes you are right about the answer choices.. one of them should be 1 and 3..


coming to your explanation, i couldnt understand how 2^18 when divided by 3 leaves a remainder 8

moreover how did you conclude from the above steps that n = 1, 4 , 7 and so on.

thanks!
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by tohellandback » Sun Aug 16, 2009 7:41 pm
clar wrote:yes you are right about the answer choices.. one of them should be 1 and 3..


coming to your explanation, i couldnt understand how 2^18 when divided by 3 leaves a remainder 8

moreover how did you conclude from the above steps that n = 1, 4 , 7 and so on.

thanks!
oops that was a typo..wrote 8 instead of 3..
anyways explanation for the remainder
2^1=2 when divided by 3 leaves remainder-2
2^2=4 when divided by 3 leaves remainder-1
.
.
all odd powers of 2 leave remainder 2
all odd powers leave remainder 1 when divided by 3
so 2^18 leaves remainder 1

for your second question
"moreover how did you conclude from the above steps that n = 1, 4 , 7 and so on. "

take a number 15. 15-3,15-(3+3),15-(3+3+3) all will be divisible by 3
take a number 17, (17-2),17-(2+3),17-(2+3+3) all will be divisible by 3

so 2^18-1, 2^18-(1+3),2^18-(1+3+3)...all will be divisible by 3
The powers of two are bloody impolite!!
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by clar » Sun Aug 16, 2009 8:14 pm
got the point...

thanks for the detailed explanation!

your effort is much appreciated !
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