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Unsociable numbers

Source: Swiss IMO TST 2016. Problem 1

July 27, 2017
number theorygreatest common divisorset

Problem Statement

Let nn be a natural number. Two numbers are called "unsociable" if their greatest common divisor is 11. The numbers {1,2,...,2n}\{1,2,...,2n\} are partitioned into nn pairs. What is the minimum number of "unsociable" pairs that are formed?
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