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.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $32$