India TST 2009 DAY 1:PROBLEM 3
Source: medium
May 14, 2009
AMCUSA(J)MOUSAMOnumber theory proposednumber theory
Problem Statement
Let be two distinct odd natural numbers.Define a Sequence like following:
a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2}).
Prove that there exists a natural number such that a_n \equal{} gcd(a,b) \forall n\ge N.