This looked like an easy exercise, when a friend of mine asked me if I know a way to prove that the decimal representation of $3^k$ always contains a zero for $k\ge k_0$, but the more I think about this question the more I find it difficult.

After some experimentation (using Mathematica), I discovered that all the numbers $3^k$ for $69\le k\le 20000$ do have a zero in there decimal representation, and that this zero does appear among the first 100 digits. This suggests that maybe $k_0=69$.

I hope that some one can help in giving some insight on this question.

almost all$n$, but all $n$ is probably really hard. If you believe the digits are random (as they "should" be), then you expect the conjecture to be true by Borel-Cantelli type arguments. $\endgroup$ – Anthony Quas Oct 10 '14 at 17:40