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 May 4th, 2010, 05:34 AM #1 Newbie   Joined: Apr 2010 From: kenya Posts: 28 Thanks: 0 How can this be solved. for an integer m>1 show that a^m is congruent to a^(m-phi(m))modulo(m) for all integers a.I tried to solve this by induction but failed.Help.
 May 4th, 2010, 05:47 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: How can this be solved. The order of a mod m divides the order of the group (this is called Lagrange's theorem), which is phi(m). Thus a^phi(m) = 1, which gives you the result you need.
 May 4th, 2010, 06:38 AM #3 Newbie   Joined: Apr 2010 From: kenya Posts: 28 Thanks: 0 Re: How can this be solved. Isn't there any elementary approach apart from using Lang range's theorem because I got this from an exercise of number theory book before the theorem was discussed.
 May 4th, 2010, 07:11 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: How can this be solved. Do you know Fermat's little theorem? It, together with the CRT, can probably be used... but that's still a lot of machinery. I'll be honest, I can't think of a more elementary method off the top of my head. What results do you have so far?
 May 4th, 2010, 07:24 AM #5 Newbie   Joined: Apr 2010 From: kenya Posts: 28 Thanks: 0 Re: How can this be solved. By induction I reached a point where I had to show that m divides (m-phi(m);1)a^(m-phi(m)-1)+..........+(m-phi(m);m-phi(m)-1)a.Am using (a;b) to mean combination of b objects from a objects.This is where I got stuck.

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