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min no of partitions of 1-100 , each group every 2 are coprime, or any 2 are not

Source: (2021 -) 2022 Dürer Math Competition Regional E4 E+1 https://artofproblemsolving.com/community/c1621671_

November 30, 2022
combinatoricsnumber theorycoprime

Problem Statement

We want to partition the integers 1,2,3,...,1001, 2, 3, . . . , 100 into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition?
We call two integers coprime if they have no common divisor greater than 11.