Remainder of Fibonacci Number Divided by Fibonacci Number is Plus or Minus Fibonacci Number

Theorem
Let $F_n$ and $F_m$ be Fibonacci numbers.

By the Division Theorem, let:
 * $F_n = q F_m + r$

where:
 * $q \in \Z$
 * $r \in \Z: 0 \le r < \size {F_m}$

Then either $r$ or $\size {F_m} - r$, or both, is a Fibonacci number.

Proof
Follows directly from Residue of Fibonacci Number Modulo Fibonacci Number.