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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

user123451241
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2 Answers2

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

Siméon
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Hint: Usually, one of the first things to try when dealing with questions of "even or odd" is to compute everything modulo 2. If that doesn't work, modulo 4 and modulo 8 are often the next thing to try.