Problem
NT-B2-M11-P004 The Base as an Unknown
#4
★★☆☆☆ Level 2 of 5
Find all integer bases \(b>6\) for which the number \(\overline{31}_b\) is divisible by \(5\).
Convert \(\overline{31}_b\) into an expression in \(b\).
We have \(\overline{31}_b=3b+1\). The condition \(5\mid3b+1\) gives \(3b\equiv-1\equiv4\pmod5\). Since \(3^{-1}\equiv2\pmod5\), we get \(b\equiv8\equiv3\pmod5\). With \(b>6\), the answer is all \(b=5t+3\), where \(t\ge1\).
Problems like this teach students not to treat base-\(b\) notation as decimal notation.