Divisibility of Fibonacci Number/Corollary

Corollary to Divisibility of Fibonacci Number

Let $F_k$ denote the $k$th Fibonacci number.

Then:

$\forall m, n \in \Z_{> 0}: F_m \divides F_{m n}$

where $\divides$ denotes divisibility.

Proof

When $m = 1$ or $n = 1$ the result is trivially true.

Otherwise, by definition of divisibility:

$m \divides m n$

and the result follows from Divisibility of Fibonacci Number.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $11$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: $(6)$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $11$