Think Simple!

Algebra Level 4

Given $100$ reals $a_{1}, a_{2},\ldots , a_{100}$ such that $\displaystyle \sum ^{100}_{i=1}a^{16}_{i} = 2^{32}$ , we have maximum value of $\displaystyle \sum ^{100}_{i=1}a^{17}_{i}$ to be of the form $a^{k}$ , where $a$ is a prime and $k$ is a positive integer. Find $(a+k)$ .