MathNet 2009-part-01 — 175 problems (EN–VI) Question 150

Let $\{x_n\}$ be a sequence such that $x_1 \in \{5, 7\}$, and $x_{n+1} \in \{5^{x_n}, 7^{x_n}\}$, for $n = 1, 2, \dots$. Determine all the possible cases of the last two digits of $x_{2009}$.