It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. This is called the strong divisibility property. However, strong divisibility does not hold for every second order sequence. In this paper we study the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization of those polynomials that satisfy the strong divisibility property. We also give formulas to calculate the greatest common divisor of those polynomials that do not satisfy the strong divisibility property.
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