Question 72
Positive integers $a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}$ satisfy $2 \leq a_{i} \leq 166$ and $a_{i}^{b_{i}} \equiv a_{i+1}^{2} (\bmod 167)$ for each $1 \leq i \leq 7$ (where $a_{8}=a_{1}$). Compute the minimum possible value of $b_{1} b_{2} \cdots b_{7}\left(b_{1}+b_{2}+\cdots+b_{7}\right)$.
Proposed by: Gregory Pylypovych
Proposed by: Gregory Pylypovych