AUSTRIAN-POLISH 2003 Problem 2

The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$ , where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$ ). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$ . For what other d is this true?