Show the given property about divisibility

Question

I want to show that if $m \geq 1, n \geq 1$ and $gcd(m,n)=1$, then $F_m F_n \mid F_{mn}$.

$F_n$ is the $n$-th Fibonacci number.

I have tried the following so far:

$$F_{mn}=F_{mn-1}+F_{mn-2}=2F_{mn-2}+F_{mn-3}=2(F_{mn-3}+F_{mn-4})+F_{mn-3}\\=3F_{mn-3}+2F_{mn-4}=\dots=\lambda F_{mn-\lambda}+(\lambda-1)F_{mn-(\lambda+1)}$$

Is it right so far? How can we continue in order to get the desired result?

Can you prove $F_m\mid F_{mn}$, $F_n\mid F_{mn}$ and $\gcd(F_m,F_n)=1$? — Angina Seng, Oct 29 '18 at 19:43
Could you give me a hint how we show these properties? @LordSharktheUnknown — Evinda, Oct 29 '18 at 19:50

Bill Dubuque · Answer 1 · 2018-10-29T19:58:15.920

0

By here the $f_k$ are a strong divisibility sequence $\,\gcd(f_j,f_k) = f_{\large \gcd(j,k)}.\,$ In particular $\,f_j\mid f_{jk}$. Thus $\,\gcd(f_m,f_n) = f_{\gcd(m,n)}\! = f_1\! = 1\,$ and $\,f_m,f_n\mid f_{mn}\Rightarrow\, {\rm lcm}(f_m,f_n) = f_m f_n\mid f_{mn}$

edited Oct 29 '18 at 19:58

answered Oct 29 '18 at 19:52

Bill Dubuque

272,048

Above we used $,\gcd(a,b)=1,\Rightarrow, {\rm lcm}(a,b) = ab.\ $ A proof is here. – Bill Dubuque Oct 29 '18 at 20:04

Show the given property about divisibility

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