Multiples in the Fibonacci Series

I found the following problem on K. Rustan M. Leino’s puzzles page: [Carroll Morgan told me this puzzle.] Prove that for any positive K, every Kth number in the Fibonacci sequence is a multiple of the Kth number in the Fibonacci sequence. More formally, for any natural number n, let F(n) denote Fibonacci number n. That is, F(0) = 0, F(1) = 1, and F(n+2) = F(n+1) + F(n).