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

Solution :
The period of $$f\left( x \right)$$  is 7. So, the period of $$f\left( {\frac{x}{3}} \right)$$   is $$\frac{7}{{\frac{1}{3}}} = 21$$
The period of $$g\left( x \right)$$  is 11. So, the period of $$g\left( {\frac{x}{5}} \right)$$   is $$\frac{{11}}{{\frac{1}{5}}} = 55$$
Hence, $${T_1} = $$  period of $$f\left( x \right)g\left( {\frac{x}{5}} \right) = 7 \times 55 = 385$$
and $${T_2} = $$  period of $$g\left( x \right)f\left( {\frac{x}{3}} \right) = 11 \times 21 = 231$$
$$\eqalign{ & \therefore {\text{ Period of }}F\left( x \right) = {\text{L}}{\text{.C}}{\text{.M}}{\text{.}}\left\{ {{T_1},\,{T_2}} \right\} \cr & = {\text{L}}{\text{.C}}{\text{.M}}{\text{.}}\left\{ {385,\,231} \right\} \cr & = 7 \times 11 \times 3 \times 5 \cr & = 1155 \cr} $$