MHB How to show that the Fibonacci sequence is a divisibility sequence?

AI Thread Summary
The discussion centers on proving that the Fibonacci sequence is a divisibility sequence, specifically using the property that the greatest common divisor of two Fibonacci numbers can be expressed as another Fibonacci number. Participants suggest utilizing the Euclidean algorithm to establish this proof. A reference to a relevant Stack Exchange post is provided for additional insights on Fibonacci modular results. The conversation emphasizes the importance of understanding the relationship between Fibonacci numbers and their divisibility properties. Overall, the focus is on finding a mathematical approach to demonstrate this divisibility sequence.
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I wanted to prove that the Fibonacci sequence is a divisibility sequence, but I don't even know how to prove it.
all I know is that $$gcd\left({F}_{m},{F}_{n}\right)={F}_{gcd\left(m,n\right)}$$ and I should somehow use the Euclidean algorithm?
 
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