MathDB

The Fibonacci sequence is defined by F_0\equal{}0, F_1\equal{}1 and F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1} for

n \ge 0

. Prove that:

(a)

The statement "F_{n\plus{}k}\minus{}F_n is divisible by

10

for all

n \in \mathbb{N}"

is true if k\equal{}60 but false for any positive integer

k<60

(b)

The statement "F_{n\plus{}t}\minus{}F_n is divisible by

100

for all

n \in \mathbb{N}"

is true if t\equal{}300 but false for any positive integer

t<300

Problem Statement