Number theory · Divisibility · Finite sets · Greatest common divisor
Let \(S\) be a finite set of natural numbers with the property that for every two elements \(x\) and \(y\) of \(S\) there exists an element \(z \in S\) such that \(z \mid x - y\). Prove that \(S\) contains an element which divides every other element of \(S\).
Does the statement remain true if \(S\) is a finite set of integers?
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Serbian Regional Competition (Okruzno takmicenje) 2023, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source