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2013 preRMO p13, a,b \in {1,2,... ,1000}, 15/(a+b)

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August 8, 2019
number theorymaximumdivides

Problem Statement

To each element of the set S={1,2,...,1000}S = \{1,2,... ,1000\} a colour is assigned. Suppose that for any two elements a,ba, b of SS, if 1515 divides a+ba + b then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
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