Given some periodic signal \(x\left ( t\right ) \) with fundamental period \(T\), and we sample it with sampling period \(T_{s}\), what condition we need so that the discrete sequence \(x\left [ n\right ] \) generated is also periodic?
The condition is that there exist some multiple of \(T_{s}\) which divides exactly some multiple of \(T\). This is the same thing as saying that frequency of the fundamental harmonic in \(x\left ( t\right ) \) over the sampling frequency is a rational number.