Multiples in the Fibonacci series

[Carroll Morgan told me this puzzle.]

Prove that for any positive K, every K^th number in the Fibonacci sequence is a multiple of the K^th 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). Prove that for any positive K and natural n, F(n*K) is a multiple of F(K).