The Fibonacci sequence

Question

a1=1,a2=1,a3=2,a4=3,a5=5,a6=8…
is defined recursively using the formulas a1=a2=1 and an+2=an+an+1 for all n≥1. Find the greatest common divisor of a484 and a2013.

Answer 1

It has been proven that

gcd(F_m,F_n) = F_gcd(m,n)

So,

gcd(F₄₈₄,F₂₀₁₃) = F_{gcd(484,2013)} = F_{11 = 89}