MathNet 2007-part-03 — 185 problems (EN–VI) Question 44

Consider the sequence $\{a_n\}_{n \ge 1}$ of positive integers with $a_n a_{n+3} = a_{n+2} a_{n+5}$ for all positive integers $n$. Determine the largest integer that always divides $\sum_{k=1}^{2550} a_{2k} a_{2k-1}$.