KöMaL 2024 K809

Let $a_1$ be a positive integer from which we create a sequence according to the following rule. If the decimal form of $a_n$ is $10A_n+b_n$ (where $b_n$ is the unit digit of $a_n$), then $a_{n+1}=A_n+6b_n$. Show that either all terms of the sequence are divisible by $59$ or none of them are.