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

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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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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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