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Number Theory Level 2

Find the last digit of 2 128 2^{128} ?

8 2 6 4

Abhisek Mohanty by Abhisek Mohanty , 21, India

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2 solutions

Nick Vandermeeren
Feb 17, 2018

I did a more rigorous solution: First I noticed that the last of a power of 2 had a pattern.

2^ 1 = 2 2^ 2 = 4 2^ 3 = 8 2^ 4 = 1 6
2^ 5 = 3 2 2^ 6 = 6 4 2^ 7 = 12 8 2^ 8 = 15 6

The pattern is in the last digit: 2, 4, 8, 6, 2, 4, 8, 6, ...

Since the exponent ends with 128 thus ends with 28 as last 2 digits I looked what the last digit was for 2^28.

So i continued the pattern as given in the table above and concluded that the power 28 would be in the 4th column thus the power would end with a 6.

2^ 1 = 2 2^ 2 = 4 2^ 3 = 8 2^ 4 = 1 6
2^ 5 = 3 2 2^ 6 = 6 4 2^ 7 = 12 8 2^ 8 = 1 5 6
2^ 9 = ... 2 2^ 10 = ... 4 2^ 11 = ... 8 2^ 12 = ... 6
2^ 13 = ... 2 2^ 14 = ... 4 2^ 15 = ... 8 2^ 16 = ... 6
2^ 17 = ... 2 2^ 18 = ... 4 2^ 19 = ... 8 2^ 20 = . .. 6
2^ 21 = ... 2 2^ 22 = ... 4 2^ 23 = ... 8 2^ 24 = ... 6
2^ 25 = ... 2 2^ 26 = ... 4 2^ 27 = ... 8 2^ 28 = ... 6
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Piyush Kumar
Aug 16, 2014

2^128 = 2^4*32 therefore, the unit digit = 2^4=6 .

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