Friday, January 19, 2018

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 f(x+y)=if(x)+jf(y) for i,jZ, so by induction f(d)f(nd)


f((n+1)d)=f(nd+d)=if(nd)+jf(d)=ikf(d)+jf(d)=(ik+j)f(d)


Remark   More generally gcd For a proof see here.


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