DE-BUNDESWETTBEWERB 2012 Problem 8

Round 2

A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge. Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?

Dataset text AI-MO AoPS. Quoted with source.

Official text: Bundeswettbewerb Mathematik 2012 official paper, page 1.

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