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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