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May 14th, 2012, 12:29 AM  #1 
Newbie Joined: May 2012 Posts: 2 Thanks: 0  what is the largest power of 2011 that divides a number
what is the largest power of 2011 that divides 
May 14th, 2012, 05:09 PM  #2 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: what is the largest power of 2011 that divides a number
Ok, I'm voting for the 0th power. 2010 is 1 mod 2011 2012 is +1 mod 2011 If n = 1 mod x, n to any even power = +1 mod x If n = +1 mod x, n to any higher power whatsoever will be +1 mod x Both 2010 and 2011 are raised to higher and specifically even powers. So the sum of the terms will be +2 mod 2011, hence not divisible by 2011. If you are not divisible by x, you can't possibly be divisible by a higher power of x. That's how I see it? Am I missing something? Last edited by skipjack; March 22nd, 2015 at 10:43 AM. 
May 14th, 2012, 06:05 PM  #3  
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: what is the largest power of 2011 that divides a number Quote:
 
May 15th, 2012, 04:28 AM  #4  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: what is the largest power of 2011 that divides a number Quote:
Last edited by skipjack; March 22nd, 2015 at 10:47 AM.  
May 15th, 2012, 04:52 AM  #5  
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: what is the largest power of 2011 that divides a number Quote:
Last edited by skipjack; March 22nd, 2015 at 10:47 AM.  
May 15th, 2012, 08:38 AM  #6 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: what is the largest power of 2011 that divides a number
I see! That does change things, no? But while goofing around with some much smaller examples, I noted that, eg, 2^5 + 4^5 is of course divisible by 3, but is not divisible by any higher power of 3. So I'm curious to see how one would go about trying to solve this question. Brute computation is out of the question for the example given! 
May 16th, 2012, 04:23 AM  #7 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361  Re: what is the largest power of 2011 that divides a number
Perhaps binomial expansion could get us somewhere. Let 2011^n be the requested power. Now I dont know if i.e. And if yes, how to proceed. Any ideas? 
March 22nd, 2015, 10:46 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,500 Thanks: 1739  

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