# Nine divisible by 9

Given any seventeen integers, show that there is at least one subset of nine integers whose sum is divisible by $9$.

(one of my friend suggest me) may be this theorem helpful

Theorem: Given any $2n-1$ integers, there is at least one subset of $n$ integers whose sum is divisible by $n$.

and also Fermat’s Little Theorem can be used

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I have known some problems of this category handled well with the pigeon hole principle. Have you given it a try? –  Aseem Dua Nov 24 '12 at 5:53
@ Aseem Dua may be.... –  UnknownController Nov 24 '12 at 6:10

Here is a generalization of the problem to which you are referring, namely the fact that given any $2n-1$ integers there exists a subset of $n$ integers divisible by $n$. Applying the theorem to the case of $n=9$ provides the desired result.