(a)  For x [n] to be periodic with fundamental period N,

 For this to be true , following condition should be satisfied ...

                                      as required to prove.

  Converse is also easy to observe.

   (b) Now if  T / T₀ = p / q ,

         fundamental period = q

         fundamental frequency = 1 / q  = 1/p (k / N)

        And thus we have :             

   (c) Now assuming T / T₀ = p / q = k / N

  Thus , p / T periods of  x( t ) are needed to obtain samples that form a single period of  x [n].