HMMT November

Let $p_{k}=\frac{128-k}{k}$, so $$\binom{127}{k}=p_{1} p_{2} \cdots p_{k}$$ Now, for $k \leq 63$, unless $32 \mid \operatorname{gcd}(k, 128-k)=\operatorname{gcd}(k, 128), p_{k} \equiv-1(\bmod 8)$. We have $p_{32}=\frac{96}{32}=3$. Thus, we have the following characterization: $$r_{k}= \begin{cases}1 & \text { if } k \text { is even and } k \leq 31 \\ 7 & \text { if } k \text { is odd and } k \leq 31 \\ 5 & \text { if } k \text { is even and } k \geq 32 \\ 3 & \text { if } k \text { is odd and } k \geq 32\end{cases}$$ We can evaluate this sum as $$\begin{aligned} 4 \cdot & (0+1+2+3+\cdots+63) \\ & +3 \cdot(-0+1-2+3-\cdots-30+31) \\ & +(32-33+34-35+\cdots+62-63) \\ = & 4 \cdot 2016+3 \cdot 16+(-16)=8064+32=8096 \end{aligned}$$