[Math] Olympiad Mathematical of Kosovo 2011 (Problem grade 9)

Question

A little boy wrote the numbers $1,2,3,…,2011$ on a blackboard. He picks any two numbers $x,y$ , erases them with a sponge and writes the number $ |x-y |$. This process continues until only one number is left. Prove that the number left is even

Answer 1

Hint: Show that, at every step of the process, the sum of the numbers on the blackboard is even.