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Black or white ??

Source: Swiss Imo Selection 2006

May 25, 2006
combinatorics proposedcombinatorics

Problem Statement

Let nn be natural number. Each of the numbers ∈{1,2,…,n}\in\{1,2,\ldots ,n\} is coloured in black or white. When we choose a number, we flip it's colour and the colour of all the numbers which have at least one common divider with the chosen number. At the beginning all the numbers were coloured white. For which nn are all the numbers black after a finite number of changes?
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