Denote the sum of all elements of set $S$ by $s(S)$. Note that set $S$ is called good if and only if $x\nmid s(S)-x\Leftrightarrow x\nmid s(S)$ for all $x\in S$. It’s clear that $1\not\in S$. Also $\{ 2,3,...,63\}$ is not a good set. Hence the maximum possible number of elements of a good set $S$ is at most $61$. Note that $\{ 1,2,...,63\} -\{ 1,4\}$ is a good set, done.

Sorry, but dont we have to prove that $\{ 1,2,...,63\} -\{ 1,4\}$ is a good set??? Because if not, then it would be a really easy exarcise since the rest are simple.

sttsmet wrote: Sorry, but dont we have to prove that $\{ 1,2,...,63\} -\{ 1,4\}$ is a good set??? Because if not, then it would be a really easy exarcise since the rest are simple. It's good because its sum, $2011$, is prime.

CT17 wrote: sttsmet wrote: Sorry, but dont we have to prove that $\{ 1,2,...,63\} -\{ 1,4\}$ is a good set??? Because if not, then it would be a really easy exarcise since the rest are simple. It's good because its sum, $2011$, is prime. Oh I understood now! Amazing solution.