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April 2nd, 2013, 02:23 AM  #1 
Member Joined: Apr 2013 Posts: 65 Thanks: 0  A Modular Arithmetic Question
What are the last 2 digits of when written in base 3?

April 2nd, 2013, 06:25 AM  #2 
Member Joined: Mar 2013 Posts: 90 Thanks: 0  Re: A modulararithmetic question and so . Hence , say . Now and . Hence , i.e. the last two digits in base 3 are 21. 
April 2nd, 2013, 06:45 AM  #3 
Member Joined: Apr 2013 Posts: 65 Thanks: 0  Re: A Modular Arithmetic Question
Oh I see. Thanks a lot, I tried to use mod 100 and then turn the result into the third base but it clearly didn't work so well. Base conversions always confuse me. I have a question though, shouldn't we use instead mod 8 instead of 9? What I mean by that is: and 
April 2nd, 2013, 07:10 AM  #4 
Member Joined: Mar 2013 Posts: 90 Thanks: 0  Re: A modulararithmetic question
We take mod 9 because we’re working in base 3 and (just as if you wanted the last two digits in base 10 you would take mod 100 as ). By the way, my original proof was incorrect in some details. I’ve edited and fixed my post. 
April 2nd, 2013, 07:33 AM  #5 
Member Joined: Apr 2013 Posts: 65 Thanks: 0  Re: A Modular Arithmetic Question
Yeah now it's clear, thanks. I meant btw, a little brain malfunction there. So we can either take mod 3 or 6 because both and , thats what I tried to mean. I have one last question. Last 2 digits of ? 
April 2nd, 2013, 11:57 AM  #6 
Member Joined: Mar 2013 Posts: 90 Thanks: 0  Re: A modulararithmetic question
Well, so try and work around that. 
April 2nd, 2013, 12:19 PM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A modulararithmetic question Quote:
 
April 2nd, 2013, 12:33 PM  #8 
Member Joined: Apr 2013 Posts: 65 Thanks: 0  Re: A Modular Arithmetic Question
CRGreathouse: I guess you missaw it, it should be 2005^(2003^(2004)+3) instead of 2005^(2003^2004) + 3. So 2003^2004 is 1 mod 6, adding 3 you get 4. 2005^(6k+4)=7 (mod 9) Nehushtan: How do you know that 9^10=1 (mod 100) ? 
April 2nd, 2013, 12:55 PM  #9  
Senior Member Joined: Mar 2012 Posts: 571 Thanks: 26  Re: A Modular Arithmetic Question Quote:
 
April 2nd, 2013, 01:11 PM  #10 
Member Joined: Apr 2013 Posts: 65 Thanks: 0  Re: A Modular Arithmetic Question
Alright thanks. I tried to take mod 40 but it didn't work out.


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