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If a continuous periodic signal is sampled with time interval Ts , and T being time period of continuous signal -- For the resultant discrete signal to be periodic following must be true-- the ratio of T/Ts = rational number = number of samples in the discrete signal.

If I take an example of sine wave with T = 300 seconds , Ts = 100 seconds Number of samples = 3

If I take an example with same sine wave T = 300 seconds , Ts = 175 seconds Number of samples = 300 / 175 = 1.714285714285..(repeating)

My question is how come the number of samples in a discrete signals is not an integer , in other words I am not able to understand how can a discrete signal have number of samples in decimals.

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There is nothing unordinary about it. Signals can have a period that is not an integer multiple of samples, and that is normal. For example, if you record a 440 Hz sine wave with a sampling rate of 48 kHz, it is not an integer multiple. It takes 11 full sine wave periods to hit an integer amount of 1200 samples.

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