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.)