On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as a,b,c such that a>gcd(b,c), then a gets replaced by a−gcd(b,c). The game ends if there is no way to denote the numbers such that a>gcd(b,c).Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers.(Theresia Eisenkölbl)
combinatoricsgreatest common divisorAustria