Question 89
Show that every number of the form
$$
2^{5^{2^{5^{\cdot^{\cdot^{\cdot}}}}}}+4^{5^{4^{5^{\cdot^{\cdot^{\cdot}}}}}}
$$
is divisible by $2008$, where the exponent towers (alternating $2,5,2,5,\ldots$ and $4,5,4,5,\ldots$ respectively) have arbitrary, mutually independent heights $\geq 3$.
$$
2^{5^{2^{5^{\cdot^{\cdot^{\cdot}}}}}}+4^{5^{4^{5^{\cdot^{\cdot^{\cdot}}}}}}
$$
is divisible by $2008$, where the exponent towers (alternating $2,5,2,5,\ldots$ and $4,5,4,5,\ldots$ respectively) have arbitrary, mutually independent heights $\geq 3$.