# Where can I get 30 other people?

A game is played with $$31$$ people.

First, each person is assigned a different integer between $$0$$ and $$30$$ inclusive. Also, an integer $$k$$ is selected such that $$0 \leq k \leq 2015$$.

Then, a computer program chooses a random subset of $$\{1, 2, 3, \dots, 2015 \}$$ with $$k$$ elements. (For $$k = 0$$, the subset is the empty set.) Each subset is equally likely to be chosen. The sum $$S$$ of the numbers in this subset is calculated.

The winner of the game is the person whose assigned number is congruent to $$S$$ modulo $$31$$.

It is given that there is one person out of all $$31$$ people that has a greater probability of winning than the other $$30$$ people. Find the sum of all possible values of $$k$$ that satisfy this.

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