Divisibility of Fibonacci numbers
Recall that Fibonacci numbers are the members of the sequence 1, 1, 2, 3, 5, 8, ..., so that every next number is the sum of two previous numbers. It is a part of the mathematical folklore that every 5th Fibonacci number is divisible by 5. It also happens that 5 is the 5th Fibonacci number. In fact, a more general statement is true:
Every n-th Fibonacci number (i.e. number [math]\displaystyle{ F_{nk} }[/math] for some k) is divisible by [math]\displaystyle{ F_n }[/math], the n-th Fibonacci number.
Among the different ways to prove that is to use the matrix
[math]\displaystyle{ A = \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix} }[/math]
It is easy to verify by induction on [math]\displaystyle{ n }[/math] that
[math]\displaystyle{ A^n = \begin{bmatrix} F_{n-1} & F_n \\ F_n & F_{n+1} \end{bmatrix} }[/math]
Using the fact that [math]\displaystyle{ A^{m+n} = A^m \times A^n }[/math] and comparing the (1,2)-entries of the two matrices, we obtain a helpful representation of [math]\displaystyle{ F_{m+n} }[/math]:
[math]\displaystyle{ F_{m+n} = F_{m-1} F_n + F_{m} F_{n+1}. }[/math]
Assuming [math]\displaystyle{ m=nk }[/math], induction and the use of the previous formula imply that [math]\displaystyle{ F_{nk} }[/math] is divisible by [math]\displaystyle{ F_n }[/math].
Recalling that two consecutive Fibonacci numbers are always relatively prime, we can establish the property of the Euclidean Algorithm. Indeed, if [math]\displaystyle{ m\gt n }[/math] are two integers such that [math]\displaystyle{ m=qn+r }[/math], [math]\displaystyle{ 0\lt r\lt n-1 }[/math] , then
[math]\displaystyle{ F_m = F_{qn+r} = F_{qn-1} F_{r}+ F_{qn} F_{r+1}, }[/math]
and therefore
[math]\displaystyle{ {\rm gcd} (F_m,F_n)= {\rm gcd}(F_n,F_r). }[/math]