Confusion regarding frequency of Discrete Time Periodic Signal While Analyzing the DFT Plot for a signal with N samples, say we find a peak in the magnitude of the DFT plot at index $k$, this implies that our signal has a high amount of similarity with an exponential phasor rotating at $\omega = (2\pi/N)k$ or equivalently a period of $N/k$ samples.
But since we are in discrete time, the fundamental period must be an integer i.e. let the fundamental period be $T$ samples
$ \implies T/m = N/k$ (where T and m are relatively prime)
Hence shouldn't the period be $Nm/k$ and hence the frequency would be $(2\pi/N)(k/m)$  ? (where $m$ is the smallest integer that makes $T$ a positive integer.)
Thank you
Your assumption that the fundamental period has to be an integer since we are in discrete time is wrong. It can be any real number, nobody says, there has to be an integer number of periods in a signal of finite length. Also, if there is no $k$ representing this exact period, the peak in the spectrum will smear across neighboring bins.