If N is a positive integer what is the remainder when 2^N

Question

If N is a positive interger what is the remainder when  2^N  is divided by 10?

1. N is divisible by 2
2. N is divisible by 4

yezz · Accepted Answer

any integer divided by 10 , the remainder is the last digit , so question asks what is the last digit for 2^n. but 2^n has a pattern when it comes to its last digit , a cycle where the last digit repeats

2^1 = 2 , 2^2 = 4 , 2^3 = 8 , 2^4 = 16 , 2^5 =32 , 2^6 = 64, ( if you look at the pattern last digit is (2,4,8,6,2,4,8,6....)

from 1

N is even , thus 2^n could end in either 4 or 6 .... insuff

from 2
N is Divisible by 4 ie 2^n = 2^4m , thus 2^4m will always ends in 6 .... suff and the last digit when 2^4m/10 is 6

B

shikha.lakhani · Answer

I have a doubt here in selection of B as an answer.
What if it is 2^4*0 it would be 2^0= 1 and the remainder would not be 6

Bunuel · Answer

The question says that If N is a positive integer, so it cannot be 0.

MT1988 · Answer

Missed to note this, if like Bunuel pointed out Positive integer wasnt mentioned i would have gotten this question wrong .. Kudos to you for thinking it..

GMATBusters · Answer

If N is a positive integer what is the remainder when  2^N  is divided by 10?
This question reduces to what is unit digit of  2^N

1. N is divisible by 2
if N = 2, unit digit of  2^N = 4, if n = 4, unit digit of  2^N  = 6. So  NOT SUFFICIENT. 
2. N is divisible by 4
If N = 4n, unit digit of  2^N  = 6. hence  SUFFICIENT.

Answer B

Calculating unit digit:
 a^(4n+r) has the same unit digit as  a^r , if exponent is multiple of 4, find unit digit of  a^4 
Principle to calculate unit digit.
1) divide the exponent by 4, and find remainder.
2) unit digit of expression will be the unit digit of base raised to power of remainder.
If remainder is 0. Raise to power of 4.

THIS METHOD IS APPLICABLE FOR ALL DIGITS,  no need to remember cyclicity for different digits.

Kinshook · Answer

Asked: If N is a positive interger what is the remainder when  2^N  is divided by 10?

1. N is divisible by 2
N = 2k 
2^N = 2^{2k} 
The remainder when  2^N  is divided by 10 = {4,6}
NOT SUFFICIENT

2. N is divisible by 4
N = 4k
2^N = 2^{4k} 
The remainder when  2^N  is divided by 10 = 6
SUFFICIENT

IMO B

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If N is a positive integer what is the remainder when 2^N

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If N is a positive integer what is the remainder when 2^N

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