Problem
NT-B1-M10-P017 Three-Digit Number in Base \(b\)
#17
★★★☆☆ Level 3 of 5
Find all bases \(b>5\) for which \((251)_b\) is divisible by \(13\).
Solve \(2b^2+5b+1\equiv0\pmod{13}\).
Checking residues \(b=0,1,\ldots,12\) gives \(b\equiv1\) or \(b\equiv3\pmod{13}\). With \(b>5\), the bases are \(b=13t+1\), \(t\ge1\), or \(b=13t+3\), \(t\ge1\).
This is no longer linear, but the modular check is small.