Digit Rearrangement and Multiples of 13

A $k$-digit multiple of 13 has its first digit moved to the end of the number. For which values of $k$ does the resulting number remain a multiple of 13?

Example: For the number 503,906, moving the first digit gives 39,065, which is still divisible by 13. However, for the number 7,969, moving the first digit results in 9,697, which is not divisible by 13.