Fibonacci sequence and divisibility.

Question

The Fibonacci sequence is defined as \begin{align} F_{0}=0, \ F_{1}=1 \end{align} and for all $n\in\mathbb{N}$ with $n\geq 2$, \begin{align} F_{n}=F_{n-2}+F_{n-1} \end{align} Show that for any positive integers $a,b$, if $a|b$, then $F_{a}|F_{b}$.

Answer 1

From your previous Q asked 2 hours before this one, for $n,a \in \Bbb N$ we have $F_{(n+1)a}= F_{na}F_{a+1}+F_{na-1}F_a$ so if $F_a$ divides $F_{na}$ then $F_a$ divides $F_{(n+1)a}.$