Question $6$: What is $gcd(n_j,n_{j+1})?($i$.$e$.,$ look at adjacent Fibonacci num$-$

bers$).$ How many steps does the algorithm take? Do Fibonacci pairs be$-$

have like your randomly chosen pairs in the previous section? What about

$gcd(n_j,n_{j+2})?$ Is there a pattern here? Try $gcd(n_j,n_{j+3}).$

Your observations should suggest propositions you can try to prove using

the definition of Fibonacci numbers

$n_{j+1}=n_j+n_{j-1}$

and the basic step

$gcd(a,b)=gcd(b,r)$

in the Euclidean algorithm. In this connection you will find yourself asking, if

you know $gcd(a,b),$ does this help to find $gcd(a,mb),$ where $m$ is an integer?

This is a question about multiplication of integers in general.