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For any $k$ element set $\{x_1, x_2, x_3, ... x_k\}$ with $x_i$ positive integers, there is a 6 element subset such that the sum of its elements is divisible by 6. What values can k take?

First I thought to let $x_i = 6m_i + l_i$ with $0\le l_i \le 5 $. Then, without loss of generality, the sum of elements of a 6 element subset would look like:

$$ 6(m_1 + m_2 + m_3 ... + m_6) + (l_1 + l_2 + l_3 ... + l_6) $$

Then I thought to use the pigeonhole principle by seeing what values $(l_1 + l_2 + l_3 ... + l_6)$ could take but it didn't lead anywhere. Any help would be appreciated.

Alp
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    Not a big help maybe, but $k\geq 12$, as for $k<12$ you can consider ${\bar 1,\bar 1,\bar 1,\bar 1,\bar 1,\bar 2,\ldots,\bar 2}$, and the rest modulo 6 of the sum of elements of any 6-elements subset is between 1 and 5. $\bar 1$ is any number of the form $1+6k$, same for $\bar2$. – Desperado Nov 16 '22 at 10:43
  • Thanks, I understand the counterexample for $k < 12$. Do you have an idea as to how it can be proven that $k \ge 12$ always works? – Alp Nov 16 '22 at 10:48
  • Sorry, my counterexample works for $k<11$, so $k\geq 11$ might work. But I have no idea how to show it, maybe you can try using a long and boring case-by-case analysis. – Desperado Nov 16 '22 at 11:12
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    See here. – Fabius Wiesner Nov 16 '22 at 13:10

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