My Math Forum the last digit

 Algebra Pre-Algebra and Basic Algebra Math Forum

 October 8th, 2013, 02:19 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, 05:17 AM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 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, 06:03 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

 Tags digit

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Albert.Teng Algebra 3 October 6th, 2012 08:37 AM horus Number Theory 22 March 24th, 2012 11:47 AM panky Algebra 0 January 16th, 2012 06:48 PM panky Algebra 8 January 10th, 2012 07:51 PM panky Algebra 3 December 13th, 2011 09:45 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top