Question 31
Let $k$ be a positive integer and $a_1, a_2, \dots, a_k$ digits. Prove that there exists a positive integer $n$ such that the last $2k$ digits of $2^n$ are, in this order, $a_1, a_2, \dots, a_k, b_1, b_2, \dots, b_k$, for some digits $b_1, b_2, \dots, b_k$.