$Theperiodoff\left(x\right)=e^{x-\left[x\right]+\left|\cos \left(\pi x\right)\right|+\left|\cos \left(2\pi x\right)\right|+\mathrm{......}\left|\cos \left(n\pi x\right)\right|}(where[.]isG.I.F)is$

$\begin{array}{l}f\left(x\right)=e^{x-\left[x\right]}+\left|\cos \left(\pi x\right)\right|+\left|\cos \left(2\pi x\right)\right|+---+\left|\cos \left(n\pi x\right)\right|\\ =e^{\left\{x\right\}}+\left|\cos \left(\pi x\right)\right|+\left|\cos \left(2\pi x\right)\right|+---+\left|\cos \left(n\pi x\right)\right|\\ Periodof\left\{x\right\}=1\\ Periodof\left|\cos \pi x\right|=\frac{\pi }{\pi }=1\\ Periodof\left|\cos 2\pi x\right|=\frac{2\pi }{\pi }=2\\ Periodof\left|\cos 3\pi x\right|=\frac{3\pi }{\pi }=3\\ :\\ :\\ Periodof\left|\mathrm{cosn}\pi x\right|=\frac{n\pi }{\pi }=\frac{1}{n}\\ \therefore Periodoff\left(x\right)=\frac{L.C.Mof(1,1,1,\mathrm{.......1})}{H.C.Fof(1,1,2,\mathrm{3....}n)}\Rightarrow \frac{1}{1}\Rightarrow 1\end{array}$