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szawinis's blog

How to prove this?
By szawinis, history, 9 years ago, In English

Given a set of n integers S, with no elements divisible by n. Prove that there exists a subset of S that has a sum divisible by n. In this case, n should be > 1.

I'm pretty trash at mathematical proofs. Can someone help me out?

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9 years ago, hide # |
 
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Arrange them in any order and look at prefix sums.

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9 years ago, hide # |
 
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You have to consider the prefix sums, but I think that the key part of the solution is knowing Pigeonhole principle

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9 years ago, hide # |
 
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Idea is like partial sum.Consider S1 = a1,S2 = a1 + a2,...Now we have n numbers (S1,S2,..Sn) and if one of them is divisible by n we're done,so asumme opposite.Now we have n number and less then n - 1 different remaniders modulo n so at least 2 of remainders (by PHP) are equal,let's say Si and Sj.But then n|Sj - Si.

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