# Not just skip counting

Let $$A = \{1, 2, 3, \dots, 2015\}.$$ Among the $$2^{2015}$$ subsets of $$A$$,

$\frac{2^p(2^q + r)}{31}$

of them have sum of elements divisible by $$31$$, for some positive integers $$p, q, r$$ such that $$r$$ is as small as possible. Find $$p + q + r \pmod{100}$$.

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