DE-BUNDESWETTBEWERB 2009 Problem 4

A positive integer is called decimal palindrome if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$ . Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.