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October 29th, 2011, 12:25 AM   #1
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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.
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October 29th, 2011, 07:34 AM   #2
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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?
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October 29th, 2011, 10:33 AM   #3
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Re: find a difference

erm..ok but what is the at least one diff example?
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October 29th, 2011, 12:45 PM   #4
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Re: find a difference

If n is 2, then by assumption both numbers are odd and hence their difference is even.
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October 29th, 2011, 06:42 PM   #5
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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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