The period of $\sin \left(\frac{2\pi x}{a}\right)+3\cos \left(\frac{2\pi x}{b}\right)$ when $a=12,b=9$ is

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\sin \left(\frac{2\mathrm{\pi x}}{\mathrm{a}}\right)+3\cos \left(\frac{2\mathrm{\pi x}}{\mathrm{b}}\right) \\ \mathrm{substitute} \mathrm{a}=12,\mathrm{b}=9 \\ \Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\sin \left(\frac{\mathrm{\pi }}{6}\mathrm{x}\right)+3\cos \left(\frac{2\mathrm{\pi }}{9}\mathrm{x}\right) \\ \mathrm{period} \mathrm{of} \sin \left(\frac{\mathrm{\pi }}{6}\mathrm{x}\right)=\frac{2\mathrm{\pi }}{{\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}=12 \\ \mathrm{period} \mathrm{of} \cos \left(\frac{2\mathrm{\pi }}{9}\mathrm{x}\right)=\frac{2\mathrm{\pi }}{{\displaystyle \raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}=9 \\ \mathrm{Period} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right)=\sin \left(\frac{2\mathrm{\pi x}}{\mathrm{a}}\right)+3\cos \left(\frac{2\mathrm{\pi x}}{\mathrm{b}}\right) \mathrm{is} \mathrm{LCM} \mathrm{of} \left\{12,9\right\}=36 \\ \boxed{\mathrm{period} =36} \end{gathered}$$