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An infinite set of integers with a certain property

Source: Baltic Way 2018, Problem 19

November 6, 2018
number theorypositive integer

Problem Statement

An infinite set BB consisting of positive integers has the following property. For each a,b∈Ba,b \in B with a>ba>b the number a−b(a,b)\frac{a-b}{(a,b)} belongs to BB. Prove that BB contains all positive integers. Here, (a,b)(a,b) is the greatest common divisor of numbers aa and bb.
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