Let Fn be the nth Fibonaaci number.

Prove the for all $\displaystyle m$ and $\displaystyle n$,

If $\displaystyle m|n$, then $\displaystyle F_m|F_n$.

I have proved three previous results which might be useful in proving the above.

For all $\displaystyle m$>= 1 and all $\displaystyle n$ >= 0 $\displaystyle F_{m+n} = F_{m-1}F_n +F_mF_{n+1}$

For all $\displaystyle m$>=1 and all $\displaystyle n$ >=1, $\displaystyle F_{m+n} = F_{m+1}F_{n+1} - F_{m-1}F_{n-1}$

$\displaystyle F_(2n+1) = (F_n)^2 + (F_{n+1})^2$ and $\displaystyle F_{2n+2} = (F_{n+2})^2 - (F_n)^2$

Thank you very much!