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    OlympiadMediumProofs7–9

    Invariant Principle

    Princípio do Invariante

    On a board, numbers 1 through 20 are written. In each step, you erase two numbers a and b, and write |a - b|. Can the last remaining number be 0?

    Consider the sum of all numbers modulo 2.

    Solution

    Step 1 of 6

    1. 1.Invariant: replacing a and b by |a - b| never changes the parity of the total sum, because |a - b| \equiv a + b \pmod{2}.