Problem
NT-B1-M10-P010 Period of \(\frac{1}{13}\)
#10
★★☆☆☆ Level 2 of 5
Find the period length of \(\frac{1}{13}\).
Find the least \(k\) such that \(10^k\equiv1\pmod{13}\).
We have \(10^2\equiv100\equiv9\pmod{13}\), \(10^3\equiv12\), \(10^4\equiv3\), \(10^5\equiv4\), \(10^6\equiv1\pmod{13}\). No smaller positive power gives \(1\), so the period is \(6\).
No decimal expansion is needed.