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Prove that the left number is even. [KMO #3, Grade 9]

Source:

March 13, 2011
invariantcombinatorics proposedcombinatorics

Problem Statement

A little boy wrote the numbers 1,2,⋯ ,20111,2,\cdots,2011 on a blackboard. He picks any two numbers x,yx,y, erases them with a sponge and writes the number ∣x−y∣|x-y|. This process continues until only one number is left. Prove that the number left is even.
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