In this question Oleg567 conjectured the following interesting fact about Fibonacci numbers: $$ k\mid F_n\quad\Longrightarrow\quad k^d\mid F_{k^{d-1}n} $$ that can be regarder as an analogue of the Hensel's lifting lemma.

I would be glad to give a proof (following the share your knowledge, Q&A-style), based on a simple combinatorial identity, and see if others come out with more elegant ones.