Question 158
Given a natural number $k$ and a sequence $\{a_n\}_{n \ge 1}$ defined as $a_1 = 2 + 2016$, $a_2 = 2^2 + 2016$, $a_3 = 2^{2^2} + 2016$, ..., $a_n = 2^{2^n} + 2016$, .... If there exist two members of the sequence that are multiples of $k$, then prove that there are infinitely many members of the sequence that are multiples of $k$.