\section*{Problem 5 - 330935}

Determine all positive integers \(n\) with the property that the three numbers \(n+1, n+10\) and \(n+55\) have a common divisor greater than 1!
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Let \(g\) be a common divisor of the three numbers. Then \(g\) is also a divisor of \((n+10)-(n+1)=9\). For the three numbers to have a common divisor greater than 1, they must all be divisible by 3. This is the case exactly for \(n=3m-1\) with any positive integer \(m\), because then \(n+1=3m, n+10=3m+9=3(m+3)\) and \(n+55=3m+54=3(m+18)\); otherwise, none of the numbers is divisible by 3, so in particular \(n+10\) and \(n+1\) are coprime, and thus all three numbers are coprime. The numbers sought are therefore those of the form \(3m-1\) with positive integers \(m\).