NT-B2-M11-P010 Three Periods

Find the order of \(10\) modulo each denominator.

For \(7\): \(10\equiv3\), \(10^2\equiv2\), \(10^3\equiv6\), and \(10^6\equiv1\pmod7\); no smaller positive power gives \(1\). The period is \(6\).

For \(13\): \(10^2\equiv9\), \(10^3\equiv12\equiv-1\pmod{13}\), so \(10^6\equiv1\), and no smaller positive power works. The period is \(6\).

For \(37\): \(10^3=1000\equiv1\pmod{37}\), while \(10\) and \(10^2\) are not congruent to \(1\). The period is \(3\).

After solving, one may ask the student to write the actual periods and compare them with the theoretical result.