Latest Google Aptitude Question SOLUTION: what is unit digit of number 2^y...where y=10^100...??

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what is unit digit of number 2^y...where y=10^100...??

Asked In Google atul pandey (11 years ago)

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SDeep Bhattacharya  (11 years ago)

y = 10^100 =(10^2)^50= (100^50)=(4*25)^50= (4^50)*(25^50)=4*(4^49)(25^50)= 4*B; Where B=(4^49)(25^50);
Now, 2^(4n)=(2^4)^n=16^n; [where n is any +ve integer]
Now, clearly thae digit of (16)^n,will always be '6',because no matter how many times we multiply 6 by 6,the last digit of the product will always remain '6'.



Now previously we have y =4*B ; where B= (4^49)(25^50),AN INTEGER;

So,(2^y)=2^(4*B)=(16)^B;

Now,Due to above mentioned arguement the last digit of {(2^y)=2^(4*B)=(16)^B}
will clearly be '6'..:-):-)

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SDeep Bhattacharya  (11 years ago)

TYPING ERROR::

In my soluiton ,in the 4th line ,I've made a typing error,the 4th line in my solution should be read as : :

"Now, clearly the unit's digit of (16)^n, will always be '6',because no matter how many times we multiply 6 by 6,the last digit of the product will always remain '6'."

Is this solution Helpfull? Yes (11) | No   

BODHISATWA CHOWDHURY  (11 years ago)

2^1=2,2^2=4,2^3=8,2^4=16,2^5=32,2^6=64,2^7=128,2^8=256. So the unit's place repeats itself after 4 numbers.10^100 is a multiple of 4 so the required number is 2^4n(where n stands for some number). So as we see all multiples of 4 like 4 and 8 have 6 in their unit's place so the answer is 6.

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varma  (11 years ago)

answer is 2
units place of 10^100 is 0
and units place of 2^1=2,2^2=4,2^3=8,2^4=6,2^5=2,2^6=4....so it repeats so for 2^10we get 2 in units place so for every multiple of 10 it is 2

Is this solution Helpfull? Yes (1) | No (16)  

nagesh  (11 years ago)

2^10=1024, so we need unit digit of 4^100, it gives 4 for odd powers (ex: 4^1=4, 4^3=64, like this) it gives 6 for even powes (ex:4^2=16, 4^4=156) so here power is 100(even power) so ans is 6

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VIBHAV ANAND  (10 years ago)

y=10^100=100000....(100 0's)
now, 2^y=2^100000....(100 0's)
also we know,
2^1=2,
2^2=4,
2^3=8,
2^4=16,
2^5=32,
2^6=64.....so on
thus,we conclude that after every 4th power of 2,unit's place digit is repeating i.e in sequence of 2,4,8,6,2,4,8,6....
and we also know that any multiple of 100 will be totally divisible by 4
thus, "y" is also divisible by 4....
therefore 2^y gives 6 at its unit place....since here power of 2 is multiple of 4,
ans=6

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Kingshuk Ghosh Dastidar  (9 years ago)

4..
2^10^100= (2^8*2^2)^100
unit digit of 2^8=6 and unit digit of 2^2=4
thus, unit digit= 4 (since 6*4=24 and unit digit is 4)
unit digit of 4^100= 4
the answer is 4

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