BxMO 2012 Problem 1

A sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ of natural numbers is defined by the rule

$$a_{n+1}=a_{n}+b_{n} (n=1,2, \ldots)$$

where $b_{n}$ is the last digit of $a_{n}$. Prove that such a sequence contains infinitely many powers of 2 if and only if $a_{1}$ is not divisible by 5 .