Pigeonholes and repunits

It is well known that any integer k has a multiple consisting of only the digits 1 and 0. As an extension of this result, we study integers of the form 111... 000 or 111 ... 111 that are a multiple of k. We show that if k > 2 and k is not a power of 3, then the multiple can be chosen to have at most k - 1 digits.