Find the period of a signal with the DTFT plot
The discrete-time Fourier transform (DTFT) can be used for general infinite length signals, and it can also be used for periodic signals if we allow Dirac delta impulses in the expression for the DTFT. You have the correspondence

$$\textrm{DTFT}\{e^{jn\omega_0}\}=2\pi\delta(\omega-\omega_0)\tag{1}$$

where it is understood that the expression $(1)$ is valid in the interval $[ -\pi,\pi)$, because the DTFT is always $2\pi$-periodic.

An $N$-periodic signal has Dirac impulses at multiples of $2\pi/N$. Note that not all of these Dirac impulses must be present, some of them can have a weight of zero. I'm sure you can take it from here.