Question 2.6.25: The positive integers m, n, m, n are written on a blackboard......

The positive integers m, n, m, n are written on a blackboard. A generalized Euclidean algorithm is applied to this quadruple as follows: if the numbers x, y, u, v appear on the board and x > y, then x-y, y, u + v, v are written instead; otherwise x, y-x, u, v + u are written instead. The algorithm stops when the numbers in the first pair become equal (they will equal the greatest common divisor of m and n). Prove that the arithmetic mean of the numbers in the second pair at that moment equals the least common multiple of m and n.