number theory - if d divides n then prove that fibonacci of d divides fibonacci of n

prove that if $d$ divides $n$ then prove that
fibonacci of $d$ divides fibonacci of $n$.

i have tried to write $F(n)$ as a multiple of $F(d)$ using the fact that $n = ad$ for some natural $a$ but got nowhere..

Answer

By the addition law $\rm\: f(x+y) = i\:f(x) + j\:f(y)\:$ for $\rm\:i,j\in \mathbb Z,\,$ so by $\,\rm\color{#c00}{induction}\,$ $\rm\: f(d)\mid f(nd)$

$$\rm f((n\!+\!1)d)\, =\, f(nd\!+\!d)\, =\, i\: \color{#c00}{f(nd)} + j\: f(d)\, =\, i\, \color{#c00}{k\, f(d)}\! + j\: f(d)\, =\, (i\,k+j)\: f(d) $$

Remark $\ $ More generally $\rm\, \gcd(f(k),f(n)) = f_{\,\gcd(k,n)}.\,$ For a proof see here.