Fibonacci divisibility proof (factors of fibonacci numbers)

I investigated the factors of fibonacci numbers and found that one factor always was a fibonacci number, for example the fibonacci number of 2 has 2 factors 1x2, and the factor 2 is a fibonacci number. Now I have to make a proof for this: I think this site explains it but I dont fully understand it: Fibonacci Numbers - Nikola- Nikolaevich Vorobšev - Google Books

Please tell me if Im on the right track, and could you please explain me on how the proof is done?

Thank you so much, your help is much appreciated