NT-B1-M10-P019 Divisibility by \(31\)

Find the order of \(10\) modulo \(31\).

Since \(31\) is coprime to \(9\), the condition \(31\mid R_n\) is equivalent to \(10^n\equiv1\pmod{31}\). Computation gives \(10^5\equiv25\), \(10^{15}\equiv1\pmod{31}\), and the smaller divisors \(1,3,5\) of \(15\) do not give \(1\). Hence the order is \(15\), so \(31\mid R_n\) exactly when \(15\mid n\).

Forces students to check minimality, not only a working exponent.