$\text{ If }f\left(x\right)\text{ and }g\left(x\right)\text{ are periodic functions with periods }7\text{ and }11,\text{ respectively, then the period of }F\left(x\right)=f\left(x\right)g\left(\frac{x}{5}\right)-g\left(x\right)f\left(\frac{x}{3}\right)\text{ is }$

$The period of f\left(x\right) is 7,so the period of f\left(\frac{x}{3}\right) is \frac{7}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}=21$$\sin ce period of f\left(x\right) = p then the period of f\left(ax+b\right)=\frac{p}{\left|a\right|}$

$$\begin{gathered} \sin ce the period of g\left(x\right)=11, the period of g\left(\frac{x}{5}\right)=\frac{11}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}=55 \\ P_1= period of f\left(x\right)·g\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right) =7 ·55=385 \end{gathered}$$

$$\begin{gathered} and P_2= period of g\left(x\right)·f\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)=11·21 =231 \\ therefore period of F\left(x\right) = LCM \left\{P_{1,}{P}_2\right\} =LCM \left\{385,231\right\} \\ = 1155 \end{gathered}$$