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 October 29th, 2011, 01:25 AM #1 Member   Joined: Sep 2011 Posts: 97 Thanks: 0 find a difference Suppose a1, a2, a3, ..., a'n' is a sequence of n integers none of which is divisible by n. Show that at least one of the differences ai ? aj , 1 ? i; j ? n and i != j, must be divisible by n.
 October 29th, 2011, 08:34 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: find a difference There are n-1 nonzero remainders after dividing by n, but you have n numbers. Thus, two of the numbers must have the same remainder after dividing by n by the Pigeonhole Principle. This solves the problem -- why?
 October 29th, 2011, 11:33 AM #3 Member   Joined: Sep 2011 Posts: 97 Thanks: 0 Re: find a difference erm..ok but what is the at least one diff example?
 October 29th, 2011, 01:45 PM #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: find a difference If n is 2, then by assumption both numbers are odd and hence their difference is even.
 October 29th, 2011, 07:42 PM #5 Member   Joined: Sep 2011 Posts: 97 Thanks: 0 Re: find a difference yea, that's what i thougjht too haha initially i thought the n must be greater than all the sequence of integers in order to make them not divisble by n, haha

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