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n=ax+by for given a,b

Source: VJIMC 2010 2.1

June 4, 2021
number theory

Problem Statement

Let aa and bb be given positive coprime integers. Then for every integer nn there exist integers x,yx,y such that n=ax+by.n=ax+by.Prove that n=abn=ab is the greatest integer for which xy≤0xy\le0 in all such representations of nn.
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