Residue of Fibonacci Number Modulo Fibonacci Number

Theorem
Let $F_n$ denote the $n$th Fibonacci number.

Then:
 * $F_{m n + r} \equiv \left({\begin{cases} F_r & : m \bmod 4 = 0 \\

\left({-1}\right)^{r + 1} F_{n - r} & : m \bmod 4 = 1 \\ \left({-1}\right)^n F_r & : m \bmod 4 = 2 \\ \left({-1}\right)^{r + 1 + n} F_{n - r} & : m \bmod 4 = 3 \end{cases} }\right) \pmod {F_n}$