Problem
NT-B1-M10-P014 When \(37\mid R_n\)
#14
★★★☆☆ Level 3 of 5
Find all \(n\ge1\) for which \(R_n\) is divisible by \(37\).
Since \(37\) is coprime to \(9\), the condition is equivalent to \(10^n\equiv1\pmod{37}\).
We have \(10^3=1000\equiv1\pmod{37}\), while \(10 ot\equiv1\) and \(10^2\equiv26 ot\equiv1\pmod{37}\). The order of \(10\) modulo \(37\) is \(3\). Hence \(37\mid R_n\) exactly when \(3\mid n\).
Shows the link between repunits and order.