October 8th, 2013, 01:19 AM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  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 
October 8th, 2013, 04:17 AM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,312 Thanks: 115  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) 
October 8th, 2013, 05:03 AM  #3 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361  Re: the last digit
You could also see that 2^44 = 16^11


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