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October 8th, 2013, 01:19 AM   #1
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the last digit

find the one's digit of 2^44
My solution:
1. 2^even power, the last digit is 4 OR 6
a. when the power is 2 x even number, the last digit is 6
b. when the power is 2 x odd number, the last digit is 4
2. 2^odd power, the last digit is 2 OR 8
a. when the power is 2(odd number) - 1, the last digit is 2
b. when the power is 2(even number) - 1, the last digit is 8

The part of the solution that applies to the problem is in boldface.
I also note that the pattern of the last digits (for increasing powers of 2) is 2, 4, 8, 6, 2, 4, 8, 6,...and so on.

My question: Is there an easier more general solution to the problem? Thanks
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October 8th, 2013, 04:17 AM   #2
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Re: the last digit

One quick way is to notice that 2^5 = 2 (mod 10). Hence 2^(5 + 4n) = 2 (mod 10), hence 2^41 = 2 (mod 10).

So, 2^44 = 6 (mod 10)

In general:

a^n (mod 10) = a^m (mod 10), where m = n (mod 4) but using 4 instead of 0.

In general, you can use the properties of the function x^n to reduce the powers in various ways.

E.g. 7^(100) = 9^(50) = 1^(25) = 1 (mod 10)
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October 8th, 2013, 05:03 AM   #3
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Re: the last digit

You could also see that 2^44 = 16^11
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